LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 

Deceived  .....MAY    5    1894         ,  i8g  -^  ^ 
^Accessions  No*5Z£~.  Class  No. 


/">"* 
V 


CARL  FRIEDRICH  GAUSS. 


L-  V      -B-r*  .    .• 

eoh,  Bng. 


MAGNETIC  MEASUREMENTS, 


WITH  AN  APPENDIX  ON  THE 


METHOD  OF  LEAST  SQUARES. 


FRANCIS  E.  NIPIIER,  A.M., 

M 

Professor  of  Physics  in  Washington  University,  President  of  the 
St.  Louis  Academy  of  Science. 


'UITI7BRSIT7' 


D.  VAN  NOSTRAND,  PUBLISHER, 
23  MURRAY  AND  27  WARREN  STREETS. 

1886. 


Library 


COPYRIGHT,  1886, 
BY  D.  VAN  NOSTRAND. 


H.    .-',    HEWITT,    PRINTER    AND    ILECTROTYPEB, 
87    ROSE    STREET,    N.    Y. 


Dept.  Mech.  Eng. 


PREFACE. 


DURING  the  last  four  or  five  years  the  writer  has 
frequently  been  requested  to  furnish  information  re- 
lating to  the  practical  details  of  a  magnetic  survey. 
The  need  of  a  brief  hand-book  to  supplement  the  in- 
structions of  the  Coast  and  Geodetic  Survey  was  felt 
by  the  writer  while  prepariug-rfor  a  similar  survey  of 
Missouri,  and  this  little  volume  is  offered  to  the  pujb- 
lic  in  the  hope  that  it  may  be  of  service  to  others 
contemplating  similar  work. 

At  the  same  time  the  great  and  growing  impor- 
tance of  electrical  and  magnetic  measurements  will 
perhaps  commend  the  volume  to  a  wider  circle  of 
readers. 

The  discussion  of  the  method  of  Least  Squares  is 
an  extension  of  an  article  in  Weisbach's  "Mechanics." 

F.  E.  N. 

ST.  Louis,  May  24,  1886. 


Bept  Mecli  Bng1. 


CONTENTS. 


PA«E 

Introduction, 7 

Declination,  ..........  11 

Determination  of  Scale  Value  of  the  Magnet,      .     -  .        .  13 

Determination  of  Scale  Value  of  Magnet  C«t    .        .  14 

Determination  of  the  Magnetic  Axis,  .        .        .        .        *  14 

Magnetic  Axis  of  (76,          .        .        .        .        •.        ;  15 

Magnetic  Declination  at  Jefferson  City,  Mo.,     .,-       .  18 

Inclination,  .          .        .        .        .        .        .        .        .        .  21 

Magnetic  Inclination  at  Jefferson  City,  Mo.,      .        .  23 

Pendulum  Vibrations,  .        .        .       .....       ^    .\  •  25 

Moment  of  Inertia,       .        .        .        . . "      .,               »        .  33 

Moment  of  Inertia  of  Ring  Y,  .        .        .  '      .        ;  36 

Moment  of  Inertia  of  the  Magnet,      .        .        .        »        «  37 

Computation  for  Moment  of  Inertia  of  Magnet  (7e,  .  40 

Moment  of  Inertia  of  Magnet  (76,       ....  41 

Correction  of  the  Oscillation  Series  for  Torsion,  .                 ,  42 

Temperature  Correction  for  Magnetic  Moment,  .        .        *  44 

Horizontal  Intensity — Oscillations,    ....  46 

Horizontal  Intensity — Computation,          ...  47 
Reduction  of  the  Time  of  Oscillation  to  that  of  an  Infinitely 

Small  Arc, 48 

Deflection  Series  for  Intensity,      ......  51 

Horizontal  Intensity — Deflections,      ....  56 

Determination  of  the  Temperature  Coefficient  q,        .        .  57 

Systems  of  Units, 65 

Explanation  of  the  Plates, 69 


C  CONTENTS. 

PAGK 

Appendix  on  the  Method  of  Least  Squares,        ...  73 

Properties  of  the  Arithmetical  Mean,        ...  74 

Observations  on  Two  or  More  Quantities,  ...  77 

Weighted  Observations,     .        .        .        •        •        •  82 

Graphical  Methods, 85 

Time  of  Elongation  of  Polaris, 90 

Table  I., 92 

Tables  II.,  III., 93 

Azimuth  of  Polaris  ut  Elongation,     .        .        .        .  94 


0? 


INTROTTOtlON. 


A  FORCE  is  completely  known  when  its  direction  and 
intensity  have  been  determined. 

The  direction  of  the  lines  of  force  in  the  earth's  gra- 
vitation field  is  indicated  by  the  plumb-line.  The 
plumb-bob  tends  to  move  downwards  along  the  line  of 
force.  The  direction  of  the  lines  of  force  of  the  earth's 
magnetic  field  would  be  indicated  by  the  direction 
taken  by  the  magnetic  axis  of  a  magnetic  needle  sus- 
pended at  its  centre  of  gravity,  so  that  it  could  move 
freely  in  any  direction.  When  thus  placed  the  opposite 
ends  of  the  magnet-needle  tend  to  move  in  opposite 
directions  along  the  line  of  force.  The  magnetic  axis  of 
a  needle  is  a  line  passing  through  its  poles,  as  will  be 
explained  more  fully  later. 

The  position  of  the  lines  of  force  is  referred  to  two 
planes — viz.,  the  horizontal  plane  and  the  plane  of  the 
geographical  meridian.  The  position  of  the  lines  of  force 
is  really  determined  by  means  of  two  magnet-needles,  one 
of  which  is  free  to  move  in  a  horizontal  plane,  and  the 
other  in  a  vertical  plane,  since  it  is  impossible  mechani- 
cally to  combine  these  motions  in  a  single  needle  of  suf- 
ficient delicacy. 

The  first  needle  determines,  at  the  point  of  observa- 
tion, the  angle  between  the  plane  of  the  geographical 
meridian  and  the  vertical  plane  containing  the  line  of 
force  at  the  point.  This  angle  is  called  the  declination. 

The  second  needle  determines  the  angle  between  the 


8  THEORY   OF   MAGNETIC   MEASUREMENTS. 

line  of  force  and  a  horizontal  plane.  This  angle  is 
called  the  inclination  or  dip.  It  remains  to  explain  the 
method  used  in  determining  the  intensity  of  force. 

In  earlier  times  the  weight  of  a  unit  mass  was  taken 
as  the  unit  of  force.  The  weight  of  a  pound  is  not  the 
same  for  different  places  on  the  earth,  and  hence  this 
unit  can  only  be  used  for  rough  work  or  for  local  deter- 
minations. 

According  to  Newton's  law  for  attraction,  the  force 
with  which  either  of  two  masses  m  and  M  attract  the 
other,  the  distance  between  them  being  d,  is 

a) 

u 

This  equation  is  verified  by  the  motions  of  the  planets 
and  of  falling  bodies.  If  M  represents  the  mass  pf  the 
earth  in  pounds  or  grammes,  m  being  the  mass  of  a  body 
at  the  surface,  d  being  the  radius  of  the  earth,  in  feet  or 
centimetres,  then  w  represents  the  force  (measured  in 
units  which  are  not  yet  supposed  to  be  fixed)  by  which 
a  spring  separating  the  two  bodies  would  be  compressed. 
This  force  is  usually  called  the  weight  of  the  mass  m. 
It  is  equally  the  weight  of  the  earth.  The  weight  of  the 
earth  is  always  equal  to  that  of  the  body  weighed,  and  is 
therefore  an  indeterminate  quantity.  £'  represents  the 
force  with  which  a  unit  mass  would  attract  another  unit 
mas*,  the  distance  between  them  being  unity. 

By  experiments  with  Att wood's  machine  it  is  shown 
that  if  a  force  acts  upon  'm  units  of  mass,  imparting  an 
acceleration  a,  the  force  is  represented  by  the  expression, 

F=Kma,  (2) 

where  TTis  the  force  which  would  impart  a  unit  accele- 
ration to  a  unit  mass,  F  being  determined  in  any  units. 


THEORY   OF   MAGNETIC   MEASUREMENTS.  9 

If  the  body  m  is  free  to  move  under  the  attraction  of 
the  earth,  it  receives  an  acceleration  g.  Hence  the  at- 
traction of  the  earth  for  the  m  units  of  mass  is 

w  =  K  mg.  (3) 

The  earth,  having  a  mass  M,  likewise  moves  with  an 
acceleration  a'  towards  the  body  m,  the  force  acting  on 
the  earth  being  therefore 

K  Ma  =  w=  Kmg, 

so  that 

,       m 

The  value  ^  is  so  small  that  a'  is  inappreciable. 

In  equation  (1)  it  is  evident  that  the  value  of  K' 
might  be  taken  as  the  unit  of  force.  The  value  K  in 
(2)  and  (3)  would  not  then  be  unity.  This  is  not  done, 
however,  but  the  unit  of  force  is  so  chosen  that  JTin  (2) 
and  (3)  is  unity.  The  unit  force  is  then  that  force 
which  can  impart  a  unit  acceleration  to  a  unit  mass 
(gramme  or  pound). 

In  centimetre-gramme  second  (abbreviated  C.  Gr.  S.) 
units  this  unit  of  force  is  called  the  dyne.  Equation  (3) 
then  becomes 

/  A  \ 

where  w  is  the  weight  of  m  units  of  mass  expressed  in 
the  above-chosen  unit.     It  also  follows  that 

w 

»-•* 

The  left  side  of  this  expression  is  the  weight  of  a  unit 
mass.  In  English  units  the  weight  of  a  pound  at  Lon- 
don is  32.1912;  at  any  other  place  the  weight  of  a 
pound  is  g  units  of  force,  where  g  is  the  acceleration  of 


10  TflllOKY   Oi1   MAGNETIC   MEASUREMENTS. 

a  falling  body  at  the  place.     In  C.  G.  S.  units  the  weight 
of  a  gramme  at  Paris  is  980.94  dynes,  and  at  any  point 

it  is  g  dynes.     The  weight  of  -  units  of  mass  is  there- 

^7 

fore  the  unit  force. 

The  weight  of  a  gramme  at  any  point  in  the  earth's 
field  gives  a  measure  of  the  force  which  acts  upon  the 
gramme  at  that  point,  tending  to  cause  motion  along 
the  lines  of  force  ;  in  other  words,  the  value  of  g  at  any 
point  is  a  measure  of  the  strength  of  the  earth's  gravita- 
tion field  at  that  point. 

Adopting  the  above  unit  of  force,  the  value  of  K'  in 
(1)  may  be  calculated  as  follows  in  C.  Gr.  S.  units :  Con- 
sider the  case  of  a  gramme  at  the  surface  of  the  earth, 
M  representing  the  mass  of  the  earth  in  grammes,  and 
d  its  mean  radius  in  centimetres.  Then  in  (1)  m  =  l; 
Jf==6.14  x  1027;  rf=6.37  X  108:  w=981.  Hence 

981  X  (6.37  X  108)2_  1 

6.14  x  1027          "1.543  x  107' 

From  this  the  mass  which  must  be  placed  at  any 
point  in  order  to  attract  an  equal  mass  with  a  force  of 
one  dyne  can  be  computed,  since 

rr,  mm 


=A/  -TT,  =  3928  grammes. 


Hence  3928  grammes  would  attract  an  equal  mass  at 
a  distance  of  one  centimetre,  with  a  force  which  would 
impart  to  one  gramme  an  acceleration  of  a  centimetre 
per  second,  or  to  3928  grammes  an  acceleration  of 

cm.  per  sec.     The  3928  grammes  is  called  the  as- 


'THEORY   OF   MAGNETIC   MEASUREMENTS.  11 

tronomical  unit  of  mass.     If  this  is  taken  as  the  unit 
mass  the  astronomical  equation  of  attraction  becomes 

mM  , 

w  =  -^->  (5) 

w  being  given  in  the  ordinary  unit  of  force — the  dyne, 
or  some  equivalent  unit. 

The  unit  magnetic  pole  or  the  electro-static  unit 
quantity  of  electricity  is  denned  in  accordance  with  this 
equation.  The  unit  pole  is  that  pole  which  will  act  upon 
an  equal  pole  at  a  unit  distance  with  a  unit  force. 


Dept.  Mecli  Bug-. 


DECLINATION. 

The  magnetic  needle  commonly  used  for  precise  de- 
terminations is  of  the  collimator  form,  consisting  of  a 
small,  cylindrical  shell  of  steel,  one  end  of  which  is 
closed  by  a  lens,  the  principal  focus  of  which  is  on  a 
scale,  etched  or  photographed  on  a  glass  plate,  which 
closes  the  other  end.  This  scale  should  be  divided  de- 
cimally into  about  100  parts,  numbered  continuously 
from  one  extremity  to  the  other.  The  angle  subtended 
at  the  middle  of  the  magnet  by  one  scale  division  should 
be  between  one  and  three  minutes.  This  angle  is  called 
the  scale  value  of  the  magnet. 

The  magnet  hangs  in  a  stirrup,  supported  on  a  long 
fibre  of  raw  silk,  its  position  in  the  stirrup  being  fixed 
by  small  brass  guide-rings  around  the  magnet.  The 
magnet  is  enclosed  in  a  box,,  from  the  top  of  which  a 
glass  tube  extends  upward,  its  top  terminating  in  a 
graduated  torsion-head  to  which  the  suspension-fibre 
is  attached.  The  ends  of  the  box  are  provided  with 


12  THEORY   OF  MAGNETIC   MEASUREMENTS. 

windows,  one  admitting  light  from  a  mirror  upon  the 
scale,  and  the  other  for  telescopic  observation.  It  is 
usually  better  to  use  well-seasoned  wood  in  the  con- 
struction of  the  box,  and  to  avoid  the  use  of  metal  in  all 
parts  nearest  to  the  magnet,  as  it  is  almost  impossible 
to  obtain  brass  free  from  iron.  It  is  necessary  to  ex- 
amine all  brass  screws  or  fittings  in  a  declinometer,  if 
they  are  near  the  magnet,  and,  if  found  to  be  magnetic, 
they  should  be  replaced  by  others,  or  proper  correction 
made  for  their  effect  on  the  position  of  the  needle  and 
on  the  intensity  of  the  field.  This  correction,  although 
really  a  function  of  the  strength  of  the  field,  may  be 
taken  as  constant. 

In  some  forms  of  instrument  the  observing  telescope 
is  connected  with  the  magnet-box,  and  mounted  with  it 
on  the  same  azimuth  circle,  the  centre  of  which  is  be- 
low the  point  of  suspension  of  the  magnet  and  in  the 
same  vertical.  In  others  it  consists  of  a  transit,  or  alt- 
azimuth instrument,  mounted  on  the  fixed  support 
which  carries  the  magnet-box.  In  the  former  case  a 
change  in  the  pointing  of  the  telescope  introduces  a 
torsion  in  the  fibre.  The  latter  instrument,  which  is 
due  to  Gauss,  is  preferable  for  field-work.] 


THEOKY   OF   MAGNETIC   MEASUREMENTS.  13 

DETERMINATION  OF  SCALE  VALUE  OF  THE 
MAGNET. 

Let  the  telescope  be  focussed  upon  the  centre  of  the 
magnet  scale,  in  which  case  we  may  assume,  in  order 
to  fix  our  ideas,  that  the  optical  axes  of  the  telescope 
and  the  collimator  magnet  coincide.  If  the  magnet 
be  turned  on  its  suspension-fibre  through  any  angle, 
and  the  telescope  be  turned  on  its  vertical  axis  through 
the  same  angle  and  in  the  same  direction,  the  optical 
axes  of  the  magnet  and  telescope  will  again  be  parallel. 
If  the  two  instruments  turned  about  a  common  vertical 
axis  the  optical  axes  will  also  be  coincident ;  but  if  they 
turn  around  parallel  axes  the  optical  axes  will  be  paral- 
lel and  not  coincident  in  the  second  position.  In  both 
cases,  however,  the  scale-reading  in  the  first  and  second 
positions  will  be  the  same.  This  preservation  of  an  un- 
changed scale-reading  in  the  case  mentioned  is  also  true 
if  the  lines  of  collimation  of  the  magnet  and  telescope 
are  not  parallel.  The  value  of  one  division  of  the  scale 
may  therefore  be  determined  by  pointing  the  telescope 
successively  on  the  principal  divisions  of  the  scale, 
taking  the  readings  of  the  azimuth  circle  for  each  point- 
ing. If  the  magnet-box  moves  with  the  telescope  the 
magnet  must  hang  on  its  suspension-fibre,  and  the 
readings  must  be  corrected  for  torsion  in  a  manner  to 
be  hereafter  explained.  If  the  magnet-box  does  not 
move  the  magnet  may  be  fastened  in  its  normal  posi- 
tion during  the  operation.  If  the  number  of  pointings 
is  odd,  the  circle-readings  corresponding  to  divisions 
equidistant  from  the  middle  of  the  scale  are  reduced  to 
the  mean  division  by  finding  their  means,  as  is  shown 
in  the  third  column  of  the  table  below.  The  mean 
reading  of  the  middle  division  is  here  338°  53'. 4. 


14 


THEOKY   OF  MAGNETIC   MEASUREMENTS. 


DETERMINATION  OF  SCALE  VALUE  OF 
MAGNET  C6. 


Azimuth  circle. 

Reading  of 

DIFFERENCE  F 

ROM  MEAN. 

Scale. 

Mean 

middle 

of  Verniers. 

division. 

Circle. 

Scale. 

160 

335°  44'.  7 

188.7 

80 

150 

336   09.0 

164.4 

70 

140 

336   34.0 

139.4 

60 

130 

336   57.5 

115.9 

50 

120 

337   19.5 

93.9 

40 

110 

337   43.0 

70.4 

30 

100 

338   06.0 

47.4 

20 

90 

338   29.5 

23.9 

10 

80 

338   52.5 

338°  52*'.  5 

00.9 

00 

70 

339   16.5 

53.0 

23.1 

10 

60 

339   39.5 

52.7 

46.1 

20 

50 

340   05.0 

54.0 

71.6 

30 

40 

340   27.5 

53.5 

94.1 

40 

30 

340   52.7 

55.1 

119.3 

50 

20 

341    15.6 

54.8 

142.2 

60 

10 

341   37.5 

53.3 

164.1 

70 

0 

342   00.0 

52.3 

186.6 

80 

338°  53'.4 

Sum  1692.0 

Sum  720 

The  scale  value  is  therefore 


DBTBEMINATION  OF  THE  MAGNETIC  AXIS. 

The  magnetic  axis  is  a  straight  line  joining  the  poles 
of  the  magnet.  If  the  magnet  is  freely  suspended  the 
axis  lies  in  the  line  of  force.  The  magnetic  axis  is  de- 
termined by  taking  scale-readings  with  the  scale  alter- 
nately erect  and  inverted.  If  the  line  of  collimation  of 
the  telescope  should  happen  to  coincide  with  the  mag- 
netic axis  of  the  magnet,  it  would  then  be  pointed  in 
the  plane  of  the  magnetic  meridian,  and  the  reading  of 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


15 


the  erect  and  the  inverted  scale  would  be  identical.  The 
division-lines  of  the  scale  should  always  be  accurately 
vertical  when  read.  The  position  of  the  magnetic  axis 
varies  continually  by  the  jarring  incident  to  travel.  A 
freshly-magnetized  magnet  is  especially  unstable,  chang- 
ing even  with  variations  of  temperature.  Declination 
magnets  should  be  carried  south  end  up,  and  should  be 
kept  away  from  other  magnets.  The  axis  should  be  de- 
termined at  each  station,  or  as  often  as  experience 
shows  to  be  necessary.  This  will,  of  course,  depend 
upon  the  age  of  the  magnet,  the  hardness  of  the  steel, 
and  the  kind  of  treatment  the  magnet  receives.  The 
custom  which  some  county  surveyors  have  of  remagnet- 
izing  their  compass-needles  whenever  they  are  dissatis- 
fied with  their  instrument  is  not  a  wise  custom. 
MAGNETIC  AXIS  OF  <76. 


Magnet. 

SCALE. 

Mean. 

Alternate 
means, 
1  and  3,  2  and 
4,  etc. 

Axis  reads. 

Left. 

Bight. 

E 
I 
E 
I 
E 
I 
E 

74.6 
79.9 
62.8 
74.9 
64.6 
70.4 
65.0 

75.1 
82.3 

87.7 
87.0 
86.1 
91.0 
86.5 

74.8 
81.1 
75.2 
80.9 
75.3 
80.7 
75.7 

7s!6o 

81.00 
75.25 

80.80 
75.50 

78.05 
78.10 
78.07 
78.05 
78.10 

78.07 

When  the  telescope  is  pointed  on  the  division  78.07 
of  the  scale  its  line  of  collimation  is  in  the  plane  of 
the  magnetic  meridian.  The  left  and  right  scale-read- 
ings in  columns  2  and  3  are  the  extreme  readings  of  the 
scale  during  an  oscillation,  it  being  assumed  that  the 
amplitude  does  not  diminish.  This  is  sufficiently  pre- 
cise for  heavy  needles  or  small  amplitudes. 


16  THEOEY  OF  MAGNETIC   MEASUREMENTS. 

The  determination  of  magnetic  declination  involves  a 
determination  of  the  direction  of  a  true  north  and 
south  line,  as  shown  by  the  azimuth  circle,  and  the 
mean  position  of  the  magnetic  axis  of  the  needle  for 
the  day,  or  for  a  series  of  days,  as  read  on  the  same 
circle.  The  daily  swing  of  the  needle  in  summer  is,  on 
the  average,  about  15',  the  north  end  of  the  needle 
being  at  its  greatest  eastern  elongation  at  about  7.15  to 
7.30  o'clock  A.M.  on  normal  summer  days,  and  at  its 
western  elongation  at  1.15  to  2  o'clock  P.M.  These 
hours  vary  somewhat  with  the  season  of  the  year  and 
for  different  parts  of  the  country.  The  mean  position 
of  the  needle,  as  deduced  from  hourly  observations 
throughout  the  day,  is,  on  the  average,  within  half  a 
minute  of  the  mean  of  eastern  and  western  elongations. 
The  mean  for  successive  days  frequently  varies  by  five 
minutes,  even  in  times  of  minimum  magnetic  disturbance. 

This  mean  position  might  be  obtained  by  pointing  on 
the  axis  reading  of  the  magnet  at  the  two  elongations, 
and  taking  the  mean  of  the  azimuth  circle-readings.  It 
is  better  to  point  approximately  on  the  axis  at  about 
6.30  A.M.,  and,  clamping  the  circle,  to  take  readings  of 
the  magnet-scale  at  intervals  of  ten  or  fifteen  minutes 
until  after  elongation  has  passed.  Leaving  the  circle 
unchanged,  if  possible,  a  similar  set  of  scale-readings 
should  be  taken,  so  as  to  include  the  afternoon  elon- 
gation. These  observations,  then,  show  any  abnormal 
changes  in  declination.  The  mean  scale-reading  of  the 
two  elongations  is  then  found  ;  if  it  should  happen  to 
coincide  with  the  magnetic  axis  the  telescope  would 
then  be  pointed  in  the  mean  magnetic  meridian  for  the 
day.  Should  it  not  thus  coincide  the  circle-reading 
must  then  be  corrected  by  the  small  angle  over  which 
the  telescope  would  sweep  in  turning  from  the  mean 


THEORY   OF   MAGNETIC    MEASUREMENTS.  17 

scale-reading  of  the  elongations  to  the  magnetic  axis. 
The  sign  of  this  correction  will  depend  upon  whether 
the  scale  is  erect  or  inverted,  whether  it  reads  from  left 
to  right  or  the  reverse  when  the  scale  is  erect,  and 
whether  the  telescope  shows  an  erect  or  an  inverted 
image.  The  silk  fihre  should  be  examined  both  before 
and  after  «ach  set  of  observations,  in  order  to  detect 
any  torsion  that  may  develop.  Changes  in  atmospheric 
humidity  are  likely  to  develop  torsion  in  a  fibre,  par- 
ticularly if  it  be  a  new  one.  The  fibre  should  be  no 
larger  than  is  necessary  to  sustain  the  magnet  without 
too  frequent  breaking.  To  examine  for  torsion,  the 
magnet  is  removed  from  the  stirrup  and  a  brass  cylin- 
der of  the  same  weight  substituted.  The  torsion-head 
should  then  be  turned  until  this  cylinder  sets  parallel  to 
the  magnet-box. 

Before  the  magnetic  observations  are  begun  the  ver- 
niers of  the  transit  should  be  set  to  0°  and  180°,  and 
the  instrument  pointed  on  some  well-defined  object  near 
the  horizon.  The  lower  clamp  being  secured,  the 
upper  one  is  released  and  the  magnetic  observations 
begun  as  explained.  The  mark  should  be  far  enough 
away  so  that  it  will  be  unnecessary  to  refocus  when  the 
telescope  is  pointed  on  the  scale.  In  choosing  a  mark 
it  is  well  to  remember  that  objects  easily  visible  in  the 
evening  may  be  invisible  in  the  morning  by  reason  of 
fogs  or  changes  in  shadows.  The  station  should  always 
be  described  by  aid  of  sketches,  and  the  distance  and 
bearing  of  corner-stones  or  other  available  points  of  re- 
ference recorded.  The  following  table  shows  the  meth- 
od of  recording  the  observations.  Specimen  blanks 
for  magnetic  observations  can  be  obtained  by  observers 
on  addressing  the  Superintendent  of  the  U.  S.  Coast 
and  Geodetic  Survey,  Washington,  D.  C. 
1* 


18 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


MAGNETIC   DECLINATION  AT  JEFFERSON 
CITY,  MO. 

In  orchard  of  Phil.  E.  Chappell.  Mark — spire  of 
State  House,  about  one  mile  distant.  Date,  Aug.  12, 
1879.  Instrument,  Declinometer  No.  3,  U.  S.  C.  and  G. 
Survey.  Magnet  No.  1,  scale  erect.  Scale  value,  1'.  90. 
Mark  reads,  A,  180°00'.0;  B,  359°58'.o  at  6  A.M. 
Line  of  detorsion,  15°.  Azimuth  circle  set  to  A  363° 
56'. 0  ;  B  183°  55'.0.  Observer,  F.  E.  N. 


Time. 

SCALE-BEADING. 

Mean. 

Remarks. 

Left. 

Right. 

A.M. 

7h18m 

81d.l 

8d.05 

83.05 

Removed  torsion  weight 

at  7h  02m  A.M. 

7  23 

83.2 

7  37 

88  '.0 

83  '.6 

83.3 

7   55 

82  .4 

84  .1 

83.3 

8   17 

82  .9 

83  .9 

83.4 

Max.  East. 

8   30 

82  .5 

83  .0 

82.75 

8   40 

, 

83.0 

9   25 

81  .9 

82  .9 

82.4 

Removed  magnet.    Line 

of  detorsion  unchang- 

ed. Replaced  magnet. 

P.M. 

Ih15m 

78d.2 

79d.O 

78.6 

Line  of  detorsion  same. 

Azimuth    circle    not 

? 

changed. 

1    25 

78  .3 

78  .9 

78.6 

1    40 

78  .2 

78  .9 

78.55 

1    52 

.  . 

.  .  . 

78.5 

Max.  West. 

2   07 

78  .2 

79  .2 

78.7 

2   16 

78.8 

2   30 

78  '.7 

79  .1 

78.9 

Line  of  detorsion,  15°. 

Mark  reads,  A 180°  Ol'.O  ;  B  360°  OO'.O. 


THEORY  OF  MAGNETIC   MEASUREMENTS.  19 

Mean  reading  E.  and  W.  elongations. .     80d.  9 
Axis  of  magnet  reads 80  .8 

Reduction  to  axis -f  0.1  =  +  0'.2 

Azimuth  circle  reads. .  A  363°  55'. 0 


Magnetic  south  reads 363    55  .2 

Mark  reads 180°  00'.  0 

Azimuth  of  mark. ,  S  175    28  .1  E 


True  south  reads 355    28.1 


Magnetic  declination  E.  of  N 8°  27'.  1 

In  illuminating  the  magnet- scale  the  direct  solar  ray 
should  never,  under  any  circumstances,  be  allowed  to 
enter  the  magnet-box.  An  illumination  from  a  white 
cloud  or  an  illuminated  sheet  of  paper  is  effective.  In 
field-work,  if  a  tent  is  not  available,  the  whole  instru- 
ment should  be  covered  with  a  soft,  heavj7  cloth,  and  the 
tripod  should  similarly  be  protected  against  solar  radia- 
tion. No  vibrations  of  the  magnet  should  be  allowed, 
excepting  small  vibrations  about  a  vertical  axis.  Larger 
vibrations  may  be  checked  by  the  end  of  the  finger  ; ,  but 
it  is  better  to  use  a  camel's-hair  brush  operated  from 
the  outside  by  means  of  a  lever  or  spring,  which  must 
be  so  arranged  that  no  air-currents  are  introduced  into 
the  magnet-box.  In  field-work  the  tripod  must  always 
be  mounted  firmly  on  stakes.  Unless  the  observations 
of  a  survey  are  made  during  intervals  of  magnetic  calms, 
it  is  necessary  to  establish  a  base  station,  where  all  ob- 
servations of  declination  as  well  as  intensity  are  observed 
at  the  same  time  as  at  the  field  station. 

The  methods  for  determining  the  true  meridian  are 
easily  accessible,  and  it  is  not  thought  necessary  to  treat 
this  part  of  the  subject.  For  the  work  of  a  survey  a 
precision  of  one  minute  of  arc  is  sufficient.  Star  obser- 


20  THEORY   OF   MAGNETIC   MEASUREMENTS. 

vations  are  the  most  satisfactory.  The  method  of  equal 
altitudes  requires  more  time  than  it  is  sometimes  conve- 
nient to  devote  to  it.  The  best  method  of  determining 
the  meridian  is  by  observation  of  a  circumpolar  star  on 
elongation. 

Table  L,  at  the  end  of  this  volume,  gives  the  time  of 
occurrence  of  the  elongation  of  the  pole-star,  correct 
within  five  minutes,  for  the  years  between  1885  and  1895. 
The  time  is  local  astronomical  time. 

Table  IV.,  at  the  close  of  this  volume,  gives  the  azimuth 
of  Polaris  at  elongation  (counted  from  the  north)  for 
the  years  1885  to  1895,  inclusive,  and  is  accurate  enough 
for  all  ordinary  purposes. 

It  should  be  observed,  however,  that  in  computing  this 
table  the  mean  declination  of  Polaris  for  the  beginning 
of  the  year  is  necessarily  used.  To  obtain  the  azimuth 
to  the  nearest  tenth  of  a  minute  the  apparent  declina- 
tion for  the  date  of  observation  should  be  employed,  and 
the  azimuth  computed  from  the  folio  wing  formula  : 

costf 
sm  A  =  -  (6) 


where  A  =  azimuth,  $  =  declination,  and  cp  =  latitude. 
The  apparent  place  of  Polaris  is  given  in  the  American 
Ephemeris  for  every  day  in  the  year.  The  difference  in 
the  mean  and  apparent  places  may  produce  a  difference 
of  O'.o  in  the  computed  azimuth,  and  values  of  the  azi- 
muth taken  from  the  table  are  therefore  subject  to  an 
error  of  that  amount. 

The  precision  with  which  the  true  meridian  must  be 
determined  depends  upon  the  precision  with  which  the 
magnetic  meridian  is  determined  —  or,  in  other  words, 
upon  the  number  of  days  of  observation. 

In  order  to  illuminate  the  field  of  the  instrument 


THEORY    OF   MAGNETIC   MEASUREMENTS.  21 

where  there  is  no  axial  illumination,  a  minute  mirror  may 
be  mounted  in  front  of  the  object-glass  of  the  telescope. 
This  mirror  may  be  mounted  on  a  ring  clasping  the  tube. 
The  position  of  the  mirror  can  be  made  adjustable  by  a 
double-jointed  rod  attached  to  the  ring.  A  bull's-eye 
lantern  will  then  serve  to  throw  light  upon  the  mirror. 
It  is,  of  course,  always  understood  that  the  observing 
telescope  is  always  kept  in  adjustment,  the  levels  and  line 
of  collimation  being  carefully  examined.  For  more  care- 
ful determinations  the  telescope  should  be  reversed  ;  but 
in  the  field-work  of  a  survey  this  is  usually  unnecessary 
if  the  instrument  is  kept  in  adjustment. 


INCLINATION. 

The  inclination  or  dip  is  the  angle  between  the  line 
of  force  and  a  horizontal  line  lying  in  the  magnetic 
meridian.  A  needle  accurately  balanced  on  a  horizon- 
tal axis  directed  in  the  magnetic  prime-vertical,  so  that 
the  needle  moves  freely  in  the  plane  of  the  magnetic 
meridian,  will,  when  magnetized,  set  with  its  magnetic 
axis  in  the  line  of  force  at  the  point.  The  position  of 
the  needle  is  determined  by  means  of  a  graduated  circle, 
having  its  plane  coincident  with  that  of  the  magnetic 
meridian.  The  zero  of  graduation  is  a  horizontal  dia- 
meter. In  some  instruments  the  ends  of  the  needle 
point  to  the  scale  divisions,  which  are  read  by  means  of 
magnifying-glasses.  In  such  instruments  the  gradua- 
tion is  not  closer  than  to  ten  minutes.  In  other  instru- 
ments the  observation  is  made  by  means  of  compound 
microscopes  having  radially-placed  threads,  which  are 
set  on  the  marked  ends  of  the  needle,  the  position  being 
determined  by  means  of  verniers. 

The  circle  of  the  instrument  is  placed  in  the  mag- 


22  THEORY   OF   MAGNETIC   MEASUREMENTS. 

netic  meridian  by  means  of  a  long,  horizontal  compass- 
needle  set  parallel  to  some  mark  drawn  on  the  box.  It 
may  also  be  done  by  taking  four  readings  of  the  azimuth 
circle  of  the  instrument,  with  the  inclination-needle 
adjusted  to  a  dip  of  90°,  as  follows  :  1st,  with  the  circle 
facing  south  (magnetic)  and  needle  (marked  side)  facing 
south  ;  2d,  circle  south,  needle  north  ;  3d,  circle  north, 
needle  south  ;  4th,  circle  north,  needle  north.  The 
mean  azimuth  circle  reading  for  these  four  positions, 
±  90°,  gives  the  circle  reading  for  the  true  meridian. 
The  line  upon  which  the  setting  compass  is  adjusted  may 
be  drawn  on  the  box  after  the  meridian  has  been  thus  de- 
termined by  the  second  method,  the  dip-needle  having, 
of  course,  been  first  removed.  The  results  of  the  two 
methods  should  be  occasionally  compared. 

The  vertical  circle  being  thus  set  in  the  magnetic 
meridian,  with  its  divided  side  east,  and  with  the  marked 
side  of  the  needle  (face)  east,  two  or  three  readings  of 
the  two  ends  of  the  needle  are  taken,  the  needle  being 
slightly  lifted  so  that  the  readings  are  independent. 
The  needle  is  then  reversed  in  its  bearings  so  that  it 
faces  west,  and  readings  are  again  taken  as  before.  The 
circle  is  then  reversed,  so  that  the  position  is  then 
"  circle  west — face  east."  After  taking  readings  in  this 
position  the  needle  is  again  reversed  in  its  bearings, 
the  position  being  "circle  west — face  west."  The 
polarity  of  the  needle  is  then  reversed,  and  the  above 
readings  are  again  taken,  beginning  with  the  last  posi- 
tion. The  table  which  follows  will  show  the  method  of 
recording  and  reducing  in  order  that  the  instrumental 
errors  may  be  investigated.  It  will  be  observed  that,  in 
the  case  shown  in  the  table,  the  centre  of  gravity  of  the 
needle  is  not  in  the  axis  of  the  needle,  but  slightly  dis- 
placed towards  the  marked  end.  The  errors  of  eccen- 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


tricity  are  corrected  by  the  reversals  described,  but  they 
should  always  be  reduced  to  a  minimum. 

MAGNETIC  INCLINATION  AT  JEFFERSON  CITY,  MO. 

Orchard  of  Phil.  E.  Chappell.    Aug.  12, 1879.   Instrument  used, 
Barrow  No.  9.    Needle  No.  3.     Observer,  F.  E.  N. 


POLARITY  OF  MARKED  END—  SOUTH. 

CIBCLE  EAST. 

CIRCLE  WEST. 

Face  east. 

Face  west. 

Face  east. 

Face  west. 

S. 

N. 

S. 

N. 

S. 

N. 

S. 

N. 

65°  56' 
42 
40 

65°  46',0 

66°  01' 
65    46 
47 

71°  02' 
70    56 
71    00 

71°  00' 
70   57 
71    00 

65°  36' 
33 
35 

65°  40' 
40 
41 

70°  50' 
71    03 
03 

70°  48' 
71    00 
71    00 

65°  51'.3 

70°  59  .3 

70°  59'.0 

65°  34'.6 

65°  40/.3 

70°  58  .6 

70°  56'.0 

65°  48'.  6 

70°  59'.2 

65°  37'.4 

70°  57'.3 

68°  23'.9 

68°  17'.4 

68°  20'.6 

POLARITY  OF  MARKED  END—  NORTH. 

CIRCLE  WEST. 

CIRCLE  EAST. 

Face  west. 

Face  east. 

Face  west. 

Face  east. 

S. 

N. 

S. 

N. 

S. 

N. 

S. 

N. 

66°  56' 
52 
47 

67°  02' 
66    59 
53 

72°  22' 
12 
15 

72°  22' 
13 
17 

66°  51' 
52 
54 

66°  58' 
59 
59 

72°  23' 
20 
22 

72°  24' 
22 
22 

66°  51'.6 

66°  58'.0 

72°  16'.3 

72°  17'.3 

66°  52'.3 

66°  58'.6 

72°  21'.6 

72°  22'.6 

66°  54'.8 

72°  16'.  8 

66°  55'.  4 

72°  22'.  1 

69°  35'.8 

69°  38'.7 

69°  37'.3 

Resulting  Inclination,  68°  58'.9 

Time  of  beginning,  11^  15  A.M.  Time  of  ending,  lli>  45"  A.M.  Magnetic 
meridian  reads  16°  24',  set  by  compass.  Left  series,  68°  59'.8.  Right  series 
68°  58'. 0. 


24  THEORY   OF   MAGNETIC    MEASUREMENTS. 

It  is  best  to  reverse  the  polarity  of  the  needle  before 
making  the  first  determination  at  any  station.     This  is 
done  by  placing  the  needle  upon  its  side  upon  a  block, 
into  which  it  fits  with  its  upper  side  nearly  flush  with 
the  surface  of  the  block.     A  hole  in  the  centre  of  the 
depression  serves  to  admit  the  axle.      Two  bar-magnets 
are  used  in  magnetizing  the  needle.     Opposite  poles  are 
brought  down  upon  the  centre  of  the  needle  on  oppo- 
site sides  of  the  axle,  the  magnets  being  inclined  to  an 
angle  of  40°  to  45°  with  the  horizontal  plane.     Preserv- 
ing this  inclination,  the   magnets   are   simultaneously 
moved  in  opposite  directions  until  they  leave  the  needle. 
The  magnets  are  then  lifted  several  inches  and  brought 
down  as  before  at  the  axle  ;  but  they  should  not  be  al- 
lowed to  touch  each  other.     The  stroke  should  be  made 
with  uniform  speed.     The  supporting  block  should  have 
a  raised  guide  along  one  side,  so  that  the  strokes  may  be 
parallel  to  the  geometrical  axis  of  the  needle,  in  order 
to  avoid  eccentricity  in  the  position  of  the  magnetic 
axis.     For  the  first  magnetization  at  any  station  three 
strokes  may  be  made  on  each  side  of  the  magnet.     For 
the  subsequent  reversal  at  the  station  four  strokes  may 
be   made.     The  magnetism   of  the  needle   diminishes 
somewhat  as  the  result  of  the  shocks  incident  to  travel, 
and  this  is  the  reason  for  the  difference  in  the  number 
of  strokes.     After  magnetization  the  needle  should  be 
at  once  placed  in  position  ;  but  it  should  not  be  used  for 
about  ten  minutes,  as  the  position  of  the  magnetic  axis 
is  likely  to  fluctuate,  and  will  give  very  discordant  re- 
sults.    In  the  table  of  reductions  it  will  be  observed 
that  the  positions  in  the  left  half  are  reproduced  in  the 
right  half,  so  that  the  two  means  should  agree,  inde- 
pendently of  the  instrumental  errors. 


THEOKY   OF   MAGNETIC    MEASUREMENTS. 


PENDULUM  VIBRATIONS. 

The  determination  of  magnetic  intensity  or  the 
strength  of  the  earth's  magnetic  field  is  made  by  a  mag- 
net which  is  oscillated  as  a  magnetic  pendulum.  It 
therefore  becomes  necessary  to  give  an  exposition  of 
pendulum  vibrations,  in  order  to  make  the  subject  intel- 
ligible. It  will  further  simplify  the  treatment  if  the 
ordinary  gravitation  pendulum  is  first  discussed,  since 
the  unit  of  force  is  usually  defined  in  terms  of  the 
weight  of  a  given  mass  of  matter.  By  (4)  the  weight 
of  a  pound  at  any  point  is  g.  This  is  the  force  on  a 
pound  at  any  point  in  the  earth's  gravitation  field.  It 
may  also  be  called  the  strength  of  the  earth's  field  at 
the  point.  In  a  study  of  the  earth's  gravitation  field 
it  is  therefore  neces- 
saryto  find  the  value 
of  g  at  a  sufficient 
number  of  points. 
The  direction  of  the 
lines  of  force  are  at 
once  indicated  by  a 
plumb-line. 

A  simple  pendu- 
lum is  a  heavy  par- 
ticle having  m  units 
of  mass,  suspended 
by  a  line  without 
mass  to  a  fixed  sup- 
port. If  deflected, 
it  tends  to  fall,  and 
is  constrained  to  slide  in  a  circle  about  the  point  of 
suspension  as  centre.  The  particle  slides  down  an  in- 
clined plane  the  angle  of  which  continually  changes, 


26  THEORY   OF   MAGNETIC   MEASUREMENTS. 

being  always  a,  where  a  is  the  angle  of  deflection. 
The  action  of  the  field  on  the  pendulum  in  a  vertical 
direction  is  w  =  mg.  The  component  along  the  path 
at  any  point  is  mg  sin  a,  the  acceleration  at  the  point 
being  g  sin  a.  The  velocity  acquired  in  falling  from  A 
to  P  is  that  due  to  MN,  the  vertical  height  of  A  above 
P  ;  hence 

v*  =  2gMN.  (7) 

Calling  L  the  radius  of  the  circle  or  the  length  of  the 
pendulum,  and  denoting  the  chord  CP  by  s  and  the 
chord  CA  by  a,  we  have,  by  similar  triangles, 


hence 
and  by  (7) 


(8) 


It  is  evident  that  v  will  have  the  same  value  whether 
the  sign  of  s  is  -f-  or  —  ;  and  for  any  one  value  of  s  there 
are  two  values  of  v,  which  are  numerically  equal  but  of 
unlike  sign.  Hence  the  velocity  at  P  and  P'  will  be  the 
same,  and  it  will  be  the  same  whether  the  ball  is  ascend- 
ing or  descending  in  its  path.  A  and  B  being  the  ex- 
tremes of  the  excursion,  the  velocity  is  zero  at  those 
points,  since  s=a.  At  G  the  velocity  is  greatest,  since 
s  =  0.  The  velocity  at  C  is 


The  time  of  vibration — that  is,  the  time  required  to 
traverse  the  arc  AB  —  can  be  easily  obtained  when  the 


THEOKY   OE   MAGNETIC   MEASUKEMEtfTS. 


arc  is  small  so  that  it  does  not  sensibly  differ  from  its 
chord. 

Thus,  let  AB 
represent  the  arc 
AGE  of  the  previ- 
ous figure.  With 
C  as  a  centre  and 
A  C,  or  0,  as  a  ra- 
dius, describe  a 
circle,  and  suppose 
a  point  to  traverse  B 
this  circle  with  an 
uniform  velocity  of 


•->/£• 


At  any  instant 
let  the  point  be  at  Q.     Its  velocity  resolved  parallel  to 
AS  will  be  Fcos  TQN=Vcos  CQP 


since  CP=s  and  CQ  =  a.  | 

This  is  the  same  velocity  which  the  pendulum  has  at  P. 
Hence  the  point  will  move  around  the  arc  AQB  in  the 
same  time  which  the  pendulum  requires  to  oscillate  from 
A  to  B.  Since  the  point  moves  over  the  arc  A  QB  =  Tta 


with  the  uniform  velocity 


of  «y 


-y,  it  follows  that  the 


time  required,  which  is  the  time  of  vibration  of  the  pen- 
dulum, is 


(10) 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


This  formula  will  give  the  time  with  very  considerable 
precision,  if  the  arc  AB  is  not  over  3°  or  4°. 

Formula  (10)  applies  directly  to  the  simple  pendulum, 
but  it  also  holds  for  the  compound  pendulum.  It,  how- 
ever, becomes  necessary  to  define  what  is  meant  by  the 
length  of  the  pendulum,  since  the  particles  which  com- 
pose it  are  not  at  equal  distances  from  the  axis  of  sus- 
pension. The  particles  nearest  to  the  axis  of  suspension 
tend  to  oscillate  in  a  shorter  time  than  those  near  the 
lower  extremity.  Hence  points  near  the  axis  have  a 
longer  time,  and  those  near  the  bottom  a  shorter  time, 
than  they  would  have  if  vibrating  alone  around  the 
same  axis.  Hence  a  series  of  points  must  exist  which 
vibrate  precisely  as  they  would  if  they  were  unconnected 
with  the  system.  These  points  are  all  in  a  straight  line 
parallel  to  the  axis  of  suspension,  and  constitute  what  is 
called  the  axis  of  oscillation. 
The  perpendicular  distance  be- 
tween the  two  axes  is  the  length 
of  the  compound  pendulum.  It 
is  the  length  of  the  simple  pen- 
dulum, which  would  make  its 
vibration  in  the  same  time. 

Let  S  and  0  represent  the 
axes  of  suspension  and  of  oscil- 
lation, G  the  centre  of  gravity 
of  the  pendulum,  and  dm  any 
small  element  of  mass. 

LetSO=L;  8G=K;  Sm=r. 
Denote  the   /_  &SC  by  6,  and 
/  mSC  by  (6  -f  a).    The  planes 
of  these  angles  are  at  right  an- 
gles to  the  axes  0  and  S,  and,  in  general,  do  not  coincide. 
The  pendulum  might  be  supposed  condensed  upon 


THEORY   OF   MAGNETIC   MEASUREMENTS.  29 

the  vertical  plane  containing  G,  and  at  right  angles  to 
the  axes  8  and  0.  If  this  change  comes  about  by 
moving  the  particles  horizontally,  a  thin  plate  will 
result  which  will  have  the  same  properties  as  the  ori- 
ginal pendulum.  The  planes  of  the  angles  6  and  6  -J-  a 
would  then  coincide. 

At  any  instant  the  linear  acceleration  of  0  is  g  sin  6. 
The  angular  acceleration  of  0  and  of  every  other  point 

of  the  system  is  y  sin  6.  If  the  element  dm  were  dis- 
connected from  the  system  its  linear  acceleration  at  this 
instant  would  be  (/sin  (6  -\-  a).  The  force  required  to 
produce  this  acceleration  on  dm  is 

F=dmgam  (6  + a).  (11) 

When  connected  with  the  system  the  real  linear  accele- 

f* 

ration  is  j  g  sin  0.  The  force  required  to  produce  this 
acceleration  is 

F'  =  dm7j-gsm6.  (12) 

The  difference  F'  —  Fis  a  force  which  must  be  applied 
to  dm  in  excess  of  its  tangential  weight  component,  in 
order  to  give  it  its  actual  acceleration  as  part  of  the 
system.  The  moment  of  this  force  about  S  is 

ra 

r  (Fr  —  F)  =  y  g  sin  6  dm  —  g  sin  (6  +  a)  dm. 

The  integral  of  this  expression  for  the  entire  pendulum 
is  necessarily  zero ;  hence 

y  sin  6  /  dmr*  =  /dmg.r  sin  (6  -j-  a). 
In  this  expression  the  integral  in  the  first  member  is  the 


30  THEORY   OJ?  MAGNETIC 

moment  of  inertia  1  of  the  entire  pendulum,  or  J/A2, 
where  A  is  the  radius  of  gyration.  The  integral  in  the 
second  member  is  the  moment  of  the  weight  of  the  entire 
pendulum,  which  is  the  same  as  though  the  whole  mass 
were  at  the  centre  of  gravity.  Hence  the  last  equation 
becomes 


L 

i  The  fact  that  #.sin  0,  the  real  linear  acceleration  of 
0,  cancels  from  this  expression,  shows  that  (13)  is  true, 
independent  of  the  position  of  the  pendulum. 

The  time  of  vibration  of  a  compound  pendulum  is 
then  found  by  substituting  this  value  of  L  in  (10),  from 
which 


<=»M/«£lF-  (14) 


The  denominator  of  (14)  is  the  moment  of  the  force 
tending  to  produce  rotation  when  the  pendulum  is  de- 
flected 90°  from  its  position  of  repose.  The  line  through 
the  centre  of  gravity,  and  at  right  angles  to  the  two 
axes,  is  then  at  right  angles  to  the  lines  of  force  of 
the  earth's  gravitation  field.  It  is  evident  that  some 
point  must  exist  in  the  pendulum,  at  which  if  its  en- 
tire mass  were  concentrated  the  moment  of  inertia 
about  $  would  remain  unchanged.  This  point  is  called 
the  centre  of  gyration,  and  its  distance  from  $  is  called 
the  radius  of  gyration.  Denoting  this  radius  by  A,  the 
value  of  /  becomes  /=  J/A2.  Hence  by  (13)  the  rela- 
tion between  A,  L,  and  K  is 

K  =  L.K,  (15) 


TttEOEY   OF  MAGNETIC 


or  the  radius  of  gyration  is  a  mean  proportional  between 
the  distances  of  the  centre  of  gravity  and  of  the  axis 
of  oscillation  from  the 
axis  S. 

For  the  simple  pen- 
dulum these  distances 
are  all  equal. 

If  the  compound  pen- 
dulum consists  of  a 
thin  rod  having  its  axis 
of  suspension  intersect- 
ing the  axis  of  figure  at 
right  angles,  the  ex- 
pression for  the  length  L  will  have  the  form 

jqa  + 

~ 

the  distances  .TTand  —K  being  the  distances  of  the  cen- 
tre of  gravity  of  the  two  parts  of  the  bar,  separated  by 
a  plane  through  8  and  at  right  angles  to  the  axis  of 
figure.  The  corresponding  radii  of  gyration  are  X  and 
—A/,  the  squares  of  both  being  positive.  If  MK=  M'K\ 
the  value  of  L,  and  hence  also  the  value  of  t,  becomes 
infinite.  This  is  the  condition  of  a  balanced  lever.  If 
the  rod  is  of  uniform  section,  this  condition  is  realized 
when  the  point  S  is  midway  between  the  extremities  of 
the  bar.  A  bar  of  steel  thus  suspended  will  not  oscillate 
as  a  gravitation  pendulum,  but  when  magnetized  it  will 
oscillate  as  a  magnetic  pendulum. 

Let  m  represent  the  quantity  of  magnetism  in  each 
end  of  the  magnet,  measured  in  the  units  already  de- 
fined (p.  11).  Let  F  represent  the  action  of  the  earth's 
magnetic  field  (or  the  earth-magnet)  on  a  unit  quantity 
of  magnetism.  Then  the  force  acting  on  each  end  of  the 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


.  5 


magnet  is  Fm.     The  points  of  application  of  this  force 
are  assumed  to  be   at  two  points  called  poles,  as  the 

points  of  application 
in  the  gravitation 
pendulum  are  assum- 
ed to  be  at  the  centres 
of  gravity.  The  poles 
are  merely  the  cen- 
tres of  the  forces  act- 
ing on  the  magnet. 
Their  position  is  not 
related  in  any  simple 
manner  to  the  geo- 
metry of  the  magnet, 
but  depends  upon  the 
law  of  magnetic  distribution.  The  distances  of  the 
poles  from  S  will  therefore  remain  undetermined,  and 
may  be  denoted  by  L.  The  force  Fm  acting  on  each 
pole  of  the  magnet  is  the  analogue  of  the  force  gM 
acting  on  the  centres  of  gravity  of  the  pendulum.  But 
in  the  magnet  the  signs  of  m  at  opposite  ends  of  the 
magnet  are  unlike,  so  that  the  moment  of  the  force 
acting  on  the  magnet  and  tending  to  produce  rotation 
when  the  magnet  lies  at  right  angles  to  the  lines  of 
force  is  not  zero,  but  2FmL,  since  the  signs  of  m  and  L 
reverse  simultaneously.  The  force  is  applied  as  a  couple. 
In  a  field  of  unit  strength  this  moment  becomes  2mL, 
which  is  usually  called  the  magnetic  moment  of  the 
magnet.  This  quantity  will  hereafter  be  denoted  by  M. 
Hence  the  time  of  vibration  of  a  magnet  free  to  oscillate 
through  the  position  of  repose  (which  is  the  direction 
of  the  lines  of  force)  becomes,  from  (14), 

~  (16) 


THEORY   OF   MAGNETIC   MEASUREMENTS.  33 

The  needle,  thus  suspended  like  a  dipping-needle, 
measures  the  total  force.  The  suspension  of  a  needle 
in  this  manner  presents  great  mechanical  difficulties. 
A  needle  hung  on  a  fibre  of  silk,  and  constrained  to 
vibrate  in  a  horizontal  plane,  is  very  much  more  sensi- 
tive. Such  a  needle  determines  the  horizontal  com- 
ponent of  the  force  F,  from  which  F  is  readily  calcu- 
lated if  the  inclination  is  known. 

For  the  horizontal  needle  the  time  of  vibration  be- 
comes 


where  H=Fcos  d,  in  which  d  is  the  angle  of  inclina- 
tion and  H  the  horizontal  component  of  the  total 
force  F. 


MOMENT  OF  INEKTIA. 

The  moment  of  inertia  /  can  be  determined  by  com- 
putation when  the  vibrating  mass  has  a  regular  geo- 
metrical form,  but  it  is  usually  better  to  use  the  indi- 
rect method  due  to  Gauss.  The  magnet  is  first  allowed 
to  vibrate  freely,  and  its  time  of  vibration,  t,  is  deter- 
mined. The  magnet  is  then  loaded  with  a  known  mass 
of  non-magnetic  substance,  so  arranged  with  reference 
to  the  axis  of  vibration  that  its  moment  of  inertia  is 
known.  Let  it  be  /'. 

In  the  first  experiment  the  time  of  vibration  is  repre- 
sented by  (17).  In  the  second  experiment  it  is  repre- 
sented by  the  equation 


«v-& 

The  observations  must  be  corrected  for  torsion,  and 


34  THEORY   OF   MAGNETIC   MEASUREMENTS. 

unless  the  temperature  of  the  two  series  is  the  same  the 
observations  for  the  free  magnet  must  be  reduced  to 
that  of  the  loaded  series.  The  method  of  making  these 
corrections  will  be  explained  later.  The  observations 
should  be  made  in  a  room  where  the  temperature  can  be 
held  uniform.  If  the  determinations  of  t  and  t'  are 
made  within  a  short  interval  of  time,  the  values  of  HM 
may  be  assumed  the  same  in  the  two  equations,  after  the 
corrections  for  torsion  and  temperature  have  been  ap- 
plied. It  is  somewhat  better  to  make  alternating  de- 
terminations of  t  and  f ,  in  order  to  eliminate  changes 
in  H.  The  two  equations  give,  when  thus  corrected, 

(19) 

It  is  most  convenient  to  add  the  moment  of  inertia 
T  in  the  shape  of  an  accurately-turned  ring  of  brass  or 
gun-metal,  the  dimensions  of  which  are  accurately  de- 
termined at  some  known  temperature.  This  ring  is 
mounted  upon  the  magnet,  the  plane  of  the  ring  being 
horizontal,  and  the  axis  of  the  ring  being  in  the  axis  of 
rotation.  This  is  accomplished  by  first  suspending  the 
magnet  in  a  horizontal  position,  and,  pointing  the  tele- 
scope on  its  scale,  the  adjustment  of  the  ring  must  be 
such  as  to  reproduce  the  same  pointing  on  the  scale. 

The  formula  for  the  moment  of  inertia  of  the  ring 
may  be  deduced  as  follows  : 

Calling  dl'  the  moment  of  inertia  of  an  elementary 
ring  of  mass  dm  of  radius  r,  thickness  h,  and  radial 
width  dr,  the  value  of  dT  is 

d,T  =  dm.  r*  =  %7rrJi.dr.D. r\ 

where  D  is  the  density  of  the  material  of  which  the  ring 
is  composed.  Integrating  between  the  limits  r'  and  rff, 


THEOKY   OF   MAGNETIC   MEASUREMENTS.  35 


1  =  2nliD  r"  r\lr  =. 

J  r' 


=  g  (»•"  +  «•")>  (20) 

where  w  is  the  mass  of  the  ring,  the  external  and  inter- 
nal radii  being  r"  and  r'.  The  following  example  will 
illustrate  the  method  of  finding  the  numerical  values  of 
/'  as  function  of  the  temperature  : 

INERTIA    RING    Y  OF   THE    U.    S.    COAST   AND 
GEODETIC   SURVEY. 

The  radii  of  the  ring  as  determined  by  Mr.  Schott 
were,  at  62°  F., 

r'  =  1.1715  inch  =0.09762  ft, 

r*=  1.4219     "      =0.11846" 

w=  812.93  grains. 

The  coefficient  of  expansion  a  was  taken  0.000,010  for 
1°  F.  For  the  centigrade  degree  it  is,  of  course,  f  of  this 
quantity. 

Denoting  the  radii  at  62°  F.  and  at  6°  F.  by  r62  and  re> 
the  values  of  the  radii  for  a  temperature  0  are 

«•'.  =  *•'„[!  +  «  (0-62)] 


and  neglecting  the  square  of  a  (6  —  62),  the  values  of 
r'     and  r"*   are 


36 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


The  errors  due  to  these  approximations  at  a  tempera- 
ture of  100°  ft  are  about  one-thirtieth  of  the  error  of 
measurement  of  the  radii,  assuming  that  the  lengths  in 
decimals  of  a  foot  are  correct  to  the  fifth  decimal  place. 

Substituting  these  values  in  equation  (20),  it  becomes 

a/j 

f»  =  I  (>•"«  +  r'\,)  [1  +  8«  (&  -  62)], 
which  for  computation  would  be  put  in  the  form 

fiiy 

I'e  -  f  ('•"»  +  «•"..)  (1  -  124«)  +  aw  (r"\,  +  r'\,)  6, 
from  which  the  following  table  has  been  computed. 
MOMENT  OF  INERTIA  OF  RING  Y. 

FOOT-GRAIN   UNITS. 


0°F 

I'e 

log.  I'Q 

60 

9.5771 

0.98123 

70 

9.5790 

0.98132 

80 

9.5810 

0.98141 

90 

9.5829 

0.98150 

100 

9.5848 

0.98158 

Degrees. 

p.  p.  log. 

1° 

1 

2 

2 

3 

3 

4 

3 

5 

4 

6 

5 

7 

6 

8 

7 

9 

8 

uepu.  iviecn.  Jung. 

THEORY   OF   MAGNETIC   MEASUREMENTS.  37 

MOMENT  OF  INERTIA  OF  THE  MAGNET. 

The  magnet  being  oscillated  at  a  temperature  6,  its 
time  of  vibration  t  becomes 


When  loaded  with  the  inertia  ring,  and  at  a  tempera- 
ture 6'  for  magnet  and  ring,  the  time  of  vibration  will  be 


The  thermometer  should  be  placed  within  the  magnet- 
box  and  read  through  a  window  in  the  same.  The  mag- 
netic moment  varies  with  the  temperature,  and  therefore 
the  value  of  t  must  be  corrected  to  the  temperature  0'. 
Both  t  and  t'  should  also  be  corrected  for  the  torsion  of 
the  suspension-fibre.  The  manner  of  making  these  cor- 
rections is  explained  later.  This  being  done,  the  two 
values  of  HM  being  assumed  equal  in  (21)  and  (22),  the 
result  is 

**& -*)-"'- 

If  6  and  &  do  not  differ  more  than  one  or  two  de- 

Ie 
grees,  the  value  of  -j-  may  be  taken  as  unity.    The  value 

of  Ifi,  is  then 


If  6'  is  the  higher  temperature,  it  will  be  observed 

that  the  effect  of  calling—  unity  is  to  make  the  result- 

•** 
ing  value  of  IQ,  too  small.      If  this  difference  is  thought 


38  THEORY   OF   MAGNETIC   MEASUREMENTS. 

to  be  appreciable,  it  may  be  assumed  that  the  deter- 

r\     i     /D/ 

mined  value  corresponds  to  a  temperature  •• —     — .  But 

/& 

it  is  very  easy  to  make  the  experiments  at  a  suffi- 
ciently constant  temperature,  so  that  no  correction  is 
needed.  This  is  best  done  by  making  the  experiments 
in  some  dry  basement-room  from  which  artificial  heat 
is  excluded,  making  use  of  temperature  fluctuations 
due  to  changes  of  temperature  of  the  external  atmo- 
sphere. The  magnet  and  ring  should  be  continu- 
ally on  or  in  the  magnetometer  with  the  thermometer, 
and  the  magnetometer-box  should  be  left  open  during 
the  intervals  between  the  observations.  In  handling 
the  ring  and  magnet  the  fingers  should  be  covered  with 
non-conducting  material,  as  rubber  or  rubber-cloth.  If 
artificial  heat  is  used  the  temperature  changes  should 
be  slow,  and  the  desired  temperature  should  be  main- 
tained constant  for  some  hours  before  the  observation, 
in  order  to  allow  the  magnetic  condition  of  the  magnet 
to  become  stable.  The  temperature  should  not  be 
raised  above  the  highest  summer  heats  (shade)  which 
are  to  be  experienced  in  summer  work. 

At  least  twenty  determinations  of  Ie  should  be  thus 
obtained  at  varying  temperatures  8.  As  the  change  in  / 
is  simply  due  to  expansion,  and  is  very  small,  the  func- 
tion may  be  considered  a  linear  one,  and  may  be  repre- 
sented by  the  equation 

Iog/=log7  +(8-8)41,  (24) 

where  log  70  is  the  value  of  log  /  at  a  temperature  6Q, 
the  mean  temperature  of  the  series,  and  z//is  the  change 
in  log  /  per  degree  of  change  in  temperature.  The 
value  of  log  /0  is  the  mean  of  the  values  log  I0  given  in 


THEORY   OF   MAGNETIC   MEASUREMENTS.  39 

the  third  column  of  the  following  table.  It  then  re- 
mains to  find  AI,  for  which  the  23  observations  furnish 
23  equations.  The  value  of  AI  may  be  determined  by 
graphical  methods.  The  simultaneous  values  of  log  IQ 
and  6  are  plotted,  and  by  aid  of  a  thread  the  position 
of  the  line  represented  by  (24)  is  determined,  the  con- 
stants for  which  are  thus  easily  determined  by  well- 
known  methods  of  analytical  geometry.  The  constants 
may  also  be  determined  by  means  of  the  method  of 
least  squares.  When  the  computations  are  properly 
arranged  this  involves  little  labor,  and  the  calculation  is 
made  in  the  table  as  an  example  of  the  method.  The 
discussion  of  this  method  is  given  in  the  appendix. 

Let  it  be  required  to  assign  a  value  to  AI  in  order  to 
most  nearly  satisfy  the  23  equations.  If  any  value  at 
random  be  assigned  to  AI,  then  in  general  the  value 
of  log  /o  -  log  I9  H-  (0  -  00)  A  I  will  not  be  zero.  If 
its  value  be  denoted  by  e,  and  if  for  convenience  we  put 


and 


there  will  be  23  equations  of  the  form 
e=y  -{-  u  AI. 

The  values  of  e  should  all  be  small  numerically,  some 
being  minus  and  some  plus.  The  sum  of  the  values  ea 
or  J2ea  for  the  twenty-three  equations  should  be  a  mini- 
mum. As  is  shown  in  the  appendix,  the  value  of  AI 
which  will  make  2e*  a  minimum  is 


u 

where  2  uy  is  the  sum  of  the  twenty-three  products  of 
the  simultaneous  values  of  u  and  y,  2u*  being  the  sum 


40 


THEORY   OF   MAGNETIC    MEASUREMENTS. 


of  the  values  u*.     This  computation  is  given  in  the  final 
columns  of  the  table.     The  resulting  value  is 


=  0.000021 
and  hence 

log  Ie  =  1.22294  -f-  0.000021  (  0-  64.8),  (25) 

from  which  the  small  table  for  the  log  moment  of  inertia 
of  the  magnet  at  various  temperatures  has  been  calcu- 
lated. 

COMPUTATION  FOR  MOMENT  OF  INERTIA  OF 
MAGNET  <76. 


Date. 

0 

Log.  Ie 

y 

u 

uy 

- 

1880-81 

Oct.     18 

68.7 

1.22326 

-0.00032 

+  3.9 

-0.00125 

15.21 

Nov.     3 

65.5 

1.22295 

+            1 

+  0.7 

+  0.00001 

0.49 

"       10 

58.4 

1.22587 

-        293 

-  6.4 

+0.01875 

40.96 

Dec.      8 

65.0 

1.22535 

-        241 

+  0.2 

-0.00048 

0.04 

"       31 

540 

1.22465 

-        171 

-10.8 

+  0.01847 

116.64 

Jan.       1 

48.5 

1.22238 

+          56 

-16.3 

-0.00913 

265.69 

2 

51.0 

1.22239 

+         55 

-13.8 

-0.00759 

190.44 

3 

53.0 

1.22227 

+          67 

-11.8 

-0.00791 

139.24 

4 

51.1 

1.22126 

+        168 

-13.7 

-0.02302 

187.69 

4 

50.4 

1.22080 

+        214 

-14.4 

-0.03082 

207.36 

'      13 

63.0 

1.22540 

-       247 

-  1.8 

+  0.00445 

3.24 

'      17 

52.0 

1,22260 

:+••        -34 

-12.8 

-0.00435 

163..84 

'       18 

48.0 

1.22135 

"+        159 

-16.8 

-0.02671 

282.24 

"       19 

53.5 

1.22248 

+          46 

-11.3 

-0.00520 

127.69 

"       26 

66.5 

1.22346 

52 

+  1-7 

-0.00088 

2.89 

"       28 

61.5 

1.22188 

+        106 

-  3.3 

-0.00350 

10.89 

Feb.      2 

61.2 

1.22241 

+         53 

-  3.6 

-0.00191 

12.96 

"        4 

65.5 

1.22223 

+          71 

+  0.7 

+  0.00050 

0.49 

5 

68.1 

1.22177 

+        117 

+  3.3 

+0.00386 

10.89 

"       23 

99.7 

1.22511 

-        217 

+  34.9 

-0.07573 

1218.01 

"       26 

103.5 

1.22224 

+          70 

+38.7 

+  0.02709 

1497.69 

March  4 

96.0 

1.22138 

+        156 

+31.2 

+  0.02867 

973.44 

"     29 

86.5 

1.22425 

-        131 

+  21.7 

-0.02843 

470.89 

Means.  . 

64.8 

1.22294 

-0.12511 

5938.92 

THEOEY   OF   MAGNETIC   MEASUREMENTS. 


41 


The  following  table  for  the  log  of  the  moment  of 
inertia  has  been  calculated  from  (25).  The  table  of 
proportional  parts  gives  the  differences  from  1°  to  9°. 
For  instance,  if  this  magnet  were  oscillated  at  a  tempe- 
rature 87°,  the  log  of  the  moment  of  inertia  of  the  mag- 
net would  be  1.22341. 

MOMENT  OF  INERTIA  OF  MAGNET  <76,  FROM  (25). 


e 

Log.  IQ 

50° 

1.22263 

60° 

1.22284 

70° 

1.22305 

80° 

1.22326 

90° 

1.22347 

100° 

1.22368 

Degrees. 

p.  p.  log. 

1 

2 

2 

4 

3 

6 

4 

8 

5 

11 

6 

13 

7 

15 

8 

17 

9 

19 

pt.  Mech 


42  THEOEY  OF  MAGNETIC   MEASUREMENTS. 

CORRECTION  OF   THE   OSCILLATION   SERIES 
FOR  TORSION. 

If  a  brass  weight  having  the  same  moment  of  inertia 
as  the  magnet  were  suspended  on  the  silk  fibre,  the 
torsion  of  the  fibre  would  cause  it  to  oscillate  slowly. 
The  time  of  vibration  would  be  expressed  by  an  equation 
of  the  form  (14)  or  (16),  viz. : 


where  #,  the  directive  constant  depends  in  this  case  upon 
the  length,  diameter,  and  material  of  the  fibre.  A 
horizontal  magnet  having  the  same  value  of  /  would, 
therefore,  have  the  time  of  vibration, 


*— »4/-wW  (3?) 


If  the  magnet  were  not  affected  by  torsion  its  time  of 
vibration  would  be 

(28) 


hence  from  (37)  and  (28) 


(29) 

It  is  here  assumed  that  the  line  of  detorsion  lies  in 
the  magnetic  meridian,  so  that  the  fibre  does  not  influ- 

ence the  position  of  the  needle  when  it  is  at  rest. 

PI 

In  order  to  find  the  value  of  -7717-,  turn  the  torsion- 

HM 

head,  to  which  the  upper  extremity  of  the  fibre  is  at- 
tached, through,  say,  90°;  the  magnet  will  follow  through 
an  angle  v,  where  v  will  be  usually  about  six  to  ten 


THEOKY  OF  MAGNETIC   MEASUREMENTS.  43 

minutes.  The  number  of  minutes  t>f  twist  in  the  fibre 
will  be  5400  —  v.  Since  v  is  small,  the  moment  of  the 
earth's  force  tending  to  bring  the  magnet  into  the 
meridian  will  be  HMv  (really  HM  sin  v)  ;  the  moment 
of  the  force  of  torsion  of  the  fibre  tending  to  deflect  the 
magnet  still  more  is  (5400  —  v)  d.  Hence  for  equili- 
brium 


or 


v 
This  in  (28)  gives,  since    TT    is  small, 


5400  +  7, 
5400 

This  gives  the  necessary  correction  when  v  is  mea- 
sured. In  determining  v  the  torsion-head  is  first  turn- 
ed 90°  in  a  +  direction,  and  then  backwards  in  a  —  di- 
rection 180°,  and  then  again  in  a  -f-  direction  90°,  which 
should  reproduce  the  original  scale-reading.  The  dif- 
ferences between  the  scale-readings  should  correspond 
to  v,  %v,  and  v,  which,  added  together  and  divided  by 
four,  gives  the  value  of  v.  If  the  twist  is  not  all  re- 
moved from  the  fibre,  the  diiferences  for  the  -[-  and 
the  —  position  will  not  be  equal.  The  observations  and 
computation  for  torsion  will  be  found  in  a  specimen 
table  of  oscillations,  p.  46. 


44  THEORY   OF  MAGNETIC   MEASUREMENTS. 


TEMPERATURE  CORRECTION  FOR  MAGNETIC 
MOMENT. 

The  magnetic  moment  of  a  magnet  diminishes  slight- 
ly for  an  increase  of  temperature,  and  the  change  may 
be  assumed  to  take  place  in  accordance  with  the  equa- 
tion: 

Xt.=.Jfl[l-q(ff-0)-]>  (31) 

where  q  is  a  very  small  quantity,  representing  the  frac- 
tion of  itself  by  which  MQ  diminishes  when  heated 
through  one  degree.  The  corrections  heretofore  dis- 
cussed having  been  made,  the  times  of  vibration  at  the 
two  temperatures  will  be 

/ — T~ 

(32) 


(33) 

Dividing  one  of  these  equations  by  the  other,  and  re- 
placing M0,  by  its  value  in  (31),  the  resulting  equation 

is  : 

which  gives  the  time  of  vibration  of  the  magnet  at  a 
temperature  6,  if  the  time  of  vibration  at  0'  is  known, 
and  in  terms  of  the  coefficient  q-.  •;  The  method  of  de- 
termining this  quantity  will  be  explained  later.  It 
should  be  observed  that  the  temperature  correction  is 
by  far  the  most  difficult  and  important  of  the  correc- 
tions to  be  made  in  magnetic  observations.  In  field- 
work  it  is  particularly  troublesome.  It  is  better  in  such 
cases  to  use  a  tent  with  double  walls  and  roof  or  with  a 


THEOEY  OF  MAGNETIC   MEASUEEMENTS.  45 

large  fly,  so  as  to  keep  the  sunlight  from  the  inner  tent. 
It  should  be  open  on  opposite  sides,  in  order  to  admit 
free  circulation  of  air,  although  it  is  not  good  to  have 
the  magnet-box  too  much  exposed  to  wind.  In  the  de- 
flection observations,  to  be  discussed  later,  the  tempera- 
ture of  the  deflecting  magnet  must  be  carefully  deter- 
mined. 
The  formula  for  oscillations  becomes,  therefore, 


=  ^->  (35) 

where 

/a  __ 


The  temperature  6  is  usually  that  of   the    deflection 
series. 

The  value  /  in  (35)  is  of  course  IQf,  or  the  moment  of 
inertia  at  the  temperature  6'  at  which  the  magnet  was 
actually  oscillated.  This  is  easily  shown  by  assuming  as 
sufficiently  precise,  the  equation 

Iv  =  /.  [1  +  c  (ff  -  (7)], 

and  giving  the  values  I9  and  I9,  to  this  quantity  in  equa- 
tions (32)  and  (33).     The  proof  easily  follows. 


.  Mech 


THEOKY  OF  MAGNETIC  MEASUKEMENTS. 


HORIZONTAL  INTENSITY-OSCILLATIONS. 

Date  Aug.  12,  1879.  Station,  Jefferson  City,  Mo.,  ChappelPs 

orchard.      Instrument,  University    Magnetometer,    Magnet   6V 

Watch,  Jiirgensen  No.  10890  ;   loses  3a  per  day.     Observers,  F. 
E.  N.  and  J.  W.  S. 


No. 
Oscill. 

Watch  reads. 

Temp. 

Extreme 
scale-readings. 

Time  of 
100  oscill. 

81° 

70d.5-90d.O 

o 

Oh  1  Qm  4.1  s  K 

10 

14    49  .7 

10osc,=688.2 

20 

15    58  .0 

68.3 

30 

17    06  .2 

68  .1 

40 

18    14  .3 

68.1 

50 

19    22  .6 

68.3 

Mean  68  .2 

100  oscill 

100 

25    04.0 

81° 

llm  228.0 

llm  228.5 

110 

26    12.2 

11    22  .5 

120 

27    20.7 

11    22  .7 

130 

28    29.0 

11    22  .8 

140 

29    37.0 

11    22.7 

150 

QH     AX.  0 

noo    a 

82° 

74d.6-86d.8 

Means            6' 

81.3 

11    22  .63 

1  scale  div.  =  2'. 345. 


Torsion 
circle. 

Scale. 

Mean. 

Diff. 

Logs. 

180° 
+  90 

74.6 
68.1 

86.8 
90.0 

80.7 
79.0 

1.7 

v  =  4'.l 
5400  +  v 

3.73272 

-180 

74.2 

91.0 

82.6 

3.6 

1.7 

5400  (a.c.) 

6.26761 

+  90 

71.9 

90.0 

80.9 

7.0 

Mean  v  =  1.75  div. 

Correction 

0.00033 

THEORY  OF  MAGNETIC   MEASUREMENTS.  47 


HORIZONTAL  INTENSITY—  COMPUTATION. 


Observed  time  of  100  oscill  .....................  682s.  63 

Time    "      1     "      ....................       6.8263 

Correction  for  rate  ....................  +0  .0002 

f  =         6  .8265 


q 

q  (6r  —  0) 
l-2(0'-0) 

0.00048 

*' 

5400  +  v' 
5400 

a.c.  /3 
/ 

MR 
M 

H 

Logs. 

+4.1 

0.83419 

1.66838 
0.00033 
9.99914 

0.00197 

0.99803 

TtU 

1.66785 

M=  0.7440 

8.33215 
0.99430 
1.22340 

0.54985 
9.87158 

0.67827 

Deflection  series,  12th  Aug.,  at  8.45  A.M. 

0  =  77.2. 

/is  from  an  old  table,  and  was  cor-          M 
rect  at  that  date  for  temp,  of  0'  =  81.3.          j=[ 

ME 
M 

9.19331 

0.54985 

9.74316 
9.87158 

48 


THEORY   OF  MAGNETIC   MEASUREMENTS. 


SEDUCTION"  OF   THE  TIME  OF  OSCILLATION 
TO  THAT  OF  AN  INFINITELY  SMALL  ARC. 

The  time  of  vibration  for  an  infinitely  small  arc  has 
already  been  shown  to  be 


HM 

The  complete  time  of  vibration  of  any  oscillating 
body  in  which  the  force  which  draws  it  towards  its  po- 
sition of  repose  is  proportional  to  the  sine  of  the  angle 
of  displacement,  is 


.) 


where  a  is  the  total  arc  described.  This  formula  or  its 
equivalent  can  be  found  in  any  good  treatise  on  analyti- 
cal mechanics. 

The  formula  may  be  written 


The  value 


where  t  is  the  observed  time  of  vibration. 
of  the  expression  within  the  parenthesis, 
denoted  by  A,  is  given  in  the  table. 

It  is  never  necessary  with  a  colli- 
mator  magnet,  or  with  one  employing  a 
reflecting  scale  according  to  the  origi- 
nal method  of  Gauss,  to  make  the 
value  of  a  over  40'  to  1°,  so  that  this 
correction  may  ordinarily  be  neglected 
altogether. 

In  case  the  correction  is  to  be  made, 
the  first  term  of  the  series  only  need 


a 

A 

0° 

0.00000 

1 

00 

2 

02 

3 

04 

4 

08 

5 

12 

6 

17 

7 

23 

8 

30 

9 

39 

10 

0.00048 

THEORY   OF   MAGNETIC   MEASUREMENTS.  49 

be  used,  and  the  sine  may  be  replaced  by  the  arc.    The 
formula  then  becomes 


<-<(•-£). 


where  a  is  the  mean  of  the  arcs  of  the  first  and  last  os- 
cillations.    An  equivalent  expression  is 


tl      aa 
^1      ~ 


where  a!  and  a"  are  the  arcs  of  the  first  and  last  oscil- 
lations. The  observations  should  be  so  arranged  that 
the  amplitude  does  not  diminish  more  than  one-third 
during  the  determination  in  order  to  apply  these  latter 
formulae. 

In  order  to  obtain  the  corrected  value  t*  in  equation 
(36),  if  correction  is  to  be  made  for  amplitude,  as  well 
as  torsion  and  temperature,  the  latter  formula  becomes 

t2 


where  the  arcs  are,  of  course,  expressed  in  circular  mea- 
sure. 

In  the  example  given  in  the  blank  the  person  observ- 
ing the  magnet-scale  called  "  time  "  on  every  tenth  oscil- 
lation, and  the  assistant  read  the  watch,  observing  the 
second-hand  by  means  of  a  magnifying-glass.  The 
time  may  also  be  taken  by  a  single  observer,  if  pro- 
vided with  a  chronometer  having  a  jumping  second  or 
half-second  hand.  The  beat  is  taken  up  and  carried  in 
mind  just  before  the  oscillation  to  be  timed  occurs. 
The  second  is  best  divided  by  noting  the  position  of  the 
middle  division  of  the  scale  at  the  beats  before  and 
after  its  transit  of  the  cross-hair.  It  is  not  necessary  to 
count  the  oscillations  between  50  and  100,  as  the  time 


50  THEORY   OF   MAGNETIC    MEASUREMENTS. 

of  occurrence  of  the  hundredth  beat  can  be  calculated, 
with  a  precision  sufficient  to  enable  one  to  recognize  it 
with  certainty,  after  the  first  half  of  the  series  is  made, 
as  is  shown  in  the  blank,  where  the  time  of  100  oscil- 
lations, calculated  from  the  first  half  of  the  series,  is 
llm  223.0,  which,  added  to  the  watch-reading  of  the 
initial  observation,  9*  13™  41". 5,  gives  9h  25m  039.5  as 
the  calculated  time  of  occurrence  of  the  hundredth  os- 
cillation. The  observation  for  time  should,  of  course, 
be  made  when  the  magnet  is  at  the  middle  of  its  swing. 
Formula  (35)  gives  the  value  of  H  in  terms  of  the  un- 
known quantity  M.  If  this  quantity  does  not  change  in 
time,  (35)  may  be  used  in  making  relative  determina- 
tions. This  method  may  be  used  with  an  old  magnet, 
if  it  is  protected  from  the  influence  of  other  magnets 
and  from  mechanical  shocks.  The  slight  decrease  in 
the  value  of  M  which  may  be  expected  may  be  de- 
tected by  oscillating  before  and  after  the  tour,  at  a  base 
station,  the  change  being  assumed  constant.  It  is, 
however,  always  better  to  determine  the  value  of  M 
fifteen  or  twenty  times  in  the  course  of  a  summer.  This 
is  done  by  obtaining  an  independent  equation  involving 
M  and  H,  as  will  now  be  shown.  This  method  is  due 
to  Gauss. 


Bept.Mecli.Biig. 


THEORY   OF   MAGNETIC    MEASUREMENTS.  51 


DEFLECTION  SERIES  FOR  INTENSITY. 

The  magnet   used  in  g         N        Eig.  6 

the  oscillation  series  is  re.  r      •<—--< -*r — ll  _ 

-m    *.m  J  -mbS 

placed  by  a  small  magnet, 

n  5,  Fig.  6.  The  oscillation  or  "intensity"  magnet, 
as  it  is  sometimes  called,  is  then  mounted  on  a  horizon- 
tal bar,  with  its  longitudinal  axis  at  right  angles  to  the 
plane  of  the  magnetic  meridian,  the  prolongation  of  its 
axis  bisecting  the  axis  of  the  suspended  needle,  as  is 
shown  in  Fig.  6.  This  causes  a  deflection  of  the  needle, 
the  angle  of  deflection  (for  a  given  distance  r  between 
the  centres  of  the  magnets)  being  greater  as  the  value 
of  H  is  less. 

In  some  magnetometers  the  telescope  and  magnet-box 
are  mounted  on  a  common  azimuth  circle,  and  the 
whole  instrument  turns  about  the  vertical  axis  of  the 
circle.  The  telescope  is  always  adjusted  on  the  central 
scale-division.  The  angle  of  deflection  is  thus  read 
on  the  azimuth  circle.  In  others  the  instrument  re- 
mains in  position  in  the  plane  of  the  magnetic  meri- 
dian, the  angle  of  deflection  being  read  on  the  magnet- 
scale.  The  latter  form  was  that  originally  devised  by 
Gauss.  These  instruments  bear  to  each  other  the  same 
relation  as  the  sine  and  the  tangent  galvanometer. 
The  discussion  for  both  forms  of  instruments  will  be 
given. 

Let  m  be  the  strength  of  thfpoleS  of  intensity  mag- 
net N  8,  21  the  distance  between' Tts  poles,  and  let  m' 
and  W  be  the  analogous  values  for  the  needle  n  s. 
Let  r  be  the  distance  between  the  centres  of  the  mag- 
nets. The  distance  I'  being  small  compared  with  r, 

the  repulsion  of  N  on  n  will  be  approximately  j—  — ^-a, 


THEOKY   OF   MAGNETIC   HEASUKEMENTS. 


and  the  attraction  of  S  on  n  will  be 


mm' 


The  differ- 


ence  between  these  expressions  will  approximately  repre- 
sent the  resultant  repulsion  of  N  S  on  n,  or 


=-  mm' 


mm 


,          1 
+..  .J 


since  higher  powers  of  —  may  be  omitted.      Hence  the 
moment  of  the  couple  acting  on  n  s  is 


p  +      ^ 
P+-  •  ) 


The  needle  is  deflected  through  a 
small  angle,  «,  and  comes  to  rest  when 
the  deflecting  moments  of  the  magnet 
N  S  and  the  earth's  field  are  equal. 
The  moment  of  the  force  due  to  the 
earth's  field  is  M'H  sin  u.  The  force 
of  N  S  on  n  s  being,  for  the  tangent 
magnetometer,  assumed  to  act  at  right 
angles  to  the  magnetic  meridian,  the 
moment  of  this  force  will  be 

21' F  cos  u. 
For  equilibrium 
M'H  sin  u 


ZMM^ 
r' 


COS  U. 


THEOKY   OF   MAGNETIC   MEASUREMENTS.  53 

This  equation  involves  several  approximations  not 
made  in  the  great  discussion  of  Gauss.  The  determi- 
nations are,  however,  subject  to  errors  of  adjustment 
and  to  errors  dependent  on  the  position  of  the  magnetic 
axes  of  the  magnets,  which  do  not  coincide  with  the 
geometrical  axes.  In  practice,  therefore,  the  value  of 

P 

2J9  in  the  small  term  2  — ,-  is  replaced  by  a  constant,  P, 

and  the  errors  of  adjustment  and  of  the  approximations 
are  thrown  upon  this  quantity.  The  equation  then  be- 
comes 

(38) 

which  is  the  same  result  as  is  obtained  by  the  more  gene- 
ral discussion  of  Gauss. 

At  each  station  the  value  of  u  is  observed  for  two 
distances,  r  and  r'.  The  two  equations  thus  obtained 

can  be  combined  by  eliminating  -==  if  the  two  determi- 
nations are  made  near  together  in  time,  and  thus  a 
value  of  P  is  obtained  which  will  ^satisfy  the  two  equa- 
tions. The  residual  errors  of  adjustment  and  of  eccen- 
tricity are  thus  thrown  upon  the  quantity  P,  which  al- 
ways turns  out  to  be  small,  if  r  and  r'  are  properly 
chosen.  The  sign  of  P  is  sometimes  -f-  and  sometimes 
— .  For  obtaining  an  average  value  of  P  it  is  custo- 
mary to  make  at  least  twenty  observations,  and  for  re- 
ducing the  observations  of  a  season  it  is  sufficient  to 
take  an  average  value  of  P  as  determined  at  all  the 
stations. 

When  the  sine  magnetometer  is  used  the  deflecting 
magnet  is  at  right  angles  to  the  needle  when  the  read- 
ing is  taken.  The  moment  of  the  deflecting  magnet  on 


54  THEORY   OF   MAGNETIC   MEASUREMENTS. 

the  needle  is  then  2l'F  instead  of  21'  F  cos  u.     Hence 
the  final  equation  becomes 

(39) 

If  the  two  deflection  series  are  made  at  different  tem- 
peratures they  must  be  reduced  to  a  common  tempera- 
ture. Let  6"  =  the  mean  temperature  of  one  series, 
and  6  that  of  the  other.  Let  u'  be  the  observed  angle 
of  deflection  in  the  first  series  :  it  is  required  to  find 
the  angle  u  of  deflection  if  the  temperature  had  been 
0,  or  that  of  the  other  series.  For  the  tangent  instru- 
ment the  two  equations  become 

M  n  r        P 

2?  =lr'  tea 

M 


p  -i 

-  -t  +  .  .  .J. 


Dividing  one  equation  by  the  other,  and  reducing  by 
equation  (31),  the  result  is 

tan  u' 
tan  „=_____.  (40) 

The  oscillation  series  is  then  also  reduced  to  a  tempe- 
rature 6,  as  has  been  already  explained. 
If  6  and  6"  do  not  differ  more  than  two  or  three  de- 

grees, the  mean  of  the  two  values   -^  unreduced  for 

temperature  may  be  taken  for  a  temperature  \(Q  +  ^")> 
to  which  the  oscillation  series  is  then  reduced. 

In  the  sine  magnetometer  no  twist  is  developed  in  the 
sustaining  fibre,  but  in  the  tangent  instrument  a  tor- 
sion correction  is  needed,  which  is  deduced  as  follows  : 

1.  The  twist  having  been  removed  from  the  fibre  by 


THEORY   OF   MAGNETIC   MEASUREMENTS.  55 

means  of  the  brass  torsion-weight,  the  small  magnet 
being  suspended,  if  the  torsion-head  is  turned  through 
90°  the  magnet  is  displaced  through  an  angle  v°.  The 
number  of  degrees  of  twist  in  the  fibre  is  90°  —  v°. 
Hence  1°  of  twist  in  the  fibre  will  displace  the  magnet 

v 
90  —  v 

2.  The  initial  conditions,  being  as  in  1,  deflect  the 
magnet  through  an  angle  u'  by  means  of  some  other 
magnet.  But  for  the  torsional  effect  of  the  fibre  the 
deflection  would  be  a  little  greater,  the  angle  being  u. 
Hence  a  twist  of  u'  degrees  in  the  fibre  causes  a  dis- 
placement of  u  —  u'  degrees  of  the  magnet,  or  a  twist 

7/    T  -  -    77  ' 

of  1°  displaces  the  magnet  ---  ;  —  degrees. 

Hence 

v  u  —  u' 

90  —  v~      u'~~ 
or 

u  —  u'  +  u'  _  v  +  90  —  v 

~~i7~  90  —  v 

and 

u__       90  1  v_ 

u'    .  90  —  v  ~  1  _  Jl  ~        ^90* 

Reducing  to  minutes, 

v    ,  ,  n 

* 


where  v  is  in  minutes.     The  torsion  correction,  there- 
fore, has  the  same  form  as  in  the  oscillation  series. 

The  form  for  the  observation  of  the  deflection  series 
is  here  given.  The  deflecting  magnet  C9  is  first  placed 
on  the  west  end  of  the  bar,  then  on  the  east  end,  etc., 
the  direction  which  the  north  end  points  being,  shown 


USI7BESIT7 

k.  »  *  0V 


56  THEORY   OF   MAGNETIC   MEASUREMENTS. 

HORIZONTAL  INTENSITY-DEFLECTIONS. 

Date  Aug.  12,  1879.  Station,  Jefferson  City  (ChappelPs). 
Magnet  C6  deflecting,  Magnet  Cn  suspended.  Instrument,  Uni- 
versity Magnetometer.  Observer,  F.  E.  N. 


,.. 

North 
end. 

Time. 

Temp. 

Scale- 
reading. 

Alternate 
Means. 

Diffs. 

r. 

m 

W 
E 
W 
E 

8h29m 
8  33 
8  36 
8  38 

75 
76 
76.5 
76 

33.0 
126.6 
33.0 
126.5 

33.00 
126.55 

93.60 
93.55 

0> 

II 

8 

34 

75.9 

93.57 

I 

E 
W 
E 
W 

8 

8 
8 
8 

30 
32 
35 
40 

75.5 

76 
76.5 

77 

125.9 
33.8 
125.8 
33.5 

125.85 
33.65 

9205 
92.15 

92.10 

8 

34 

76.2 

Means          76.0 

2u 

92.835 

Computation,     —  =  |r3  tan  u  (1  -  J. 

Torsion              Scale, 
circle. 

Mean. 

Diffs. 

u  =  46.417 
ldiv.=  2'.89 
5400  +  v 
5400 

u  =  134'  .43  ) 
—  2°  14'  43  j" 

Logs. 

1.66668 
0.46090 

0.00091 

180 
270 
90 
180 

79.7 
83.6 

75.7 
79.7 

3.9 
7.9 
4.0 

79.9     *79.5 

2.12849 

Met 

3.95 

in  v  = 

tan  u 
rz 

M 
H 

8.59241 
0.90309 
9.69897 

9.99887 
9.19334 

5400  +'v 
5400  (a.c.) 

Logs. 

3.73330 
6.26761 

Correction 

0.00091 

THEORY   OP  MAGNETIC   MEASUREMENTS.  57 

in  the  second  column.  The  order  of  the  experiments 
is  indicated  in  the  time  column,  and  is  so  arranged  as 
to  eliminate  the  hourly  change  in  declination.  The 
method  of  reduction  will  be  sufficiently  apparent  by  in- 
spection of  the  blank.  A  second  series  with  r  =  1.75 
ft.  at  a  mean  temperature  6  of  78.4  gave  for  a  corrected 
value  of  u  3°  20'.  66.  These  two  series  were  used  in 
calculating  the  value  of  P  in  equation  (38),  which  was 
found  to  be  -f  0.0083.  The  value  of  P  used  in  the  final 
reduction  is,  however,  a  mean  for  the  work  of  a  season. 

The  resulting  value  of  log  TV  for  the  second  position  was 
9.19326.  Hence  for  the  two  series  the  mean  tempera- 
ture 6  is  77°.2,and  the  mean  value  ^is  9.19331,  which 

JJ. 

are  used  in  the  oscillation  blank. 


DETERMINATION  OF  THE  TEMPERATURE 
COEFFICIENT  q. 

The  temperature  coefficient  has  been  used  in  correct- 
ing both  the  oscillation  and  the  deflection  series.  Either 
of  these  correction  formulae  may  be  used  in  determin- 
ing the  value  of  q  if  all  the  other  values  in  the  equation 
are  directly  observed.  If  the  oscillation  series  is  used 
the  magnet-box  may  be  surrounded  by  a  copper  water- 
jacket,  with  windows,  closed  with  double  walls  of  mica 
or  glass,  to  admit  of  the  proper  illumination  and  to  en- 
able one  to  read  the  thermometer,  which  should  be 
placed  within  the  magnet-box.  The  copper  vessel  must 
be  first  examined  to  see  if  it  acts  magnetically,  and,  if 
so,  its  position  must  remain  unchanged  during  the  de- 
termination. The  jacket  is  first  filled  with  ice-water, 
3 


58  THEORY  OF  MAGNETIC   MEASUREMENTS. 

and  ice-water  is  allowed  to  slowly  run  through  the 
jacket  in  such  a  manner  as  to  secure  as  uniform  a  cir- 
culation as  possible.  The  temperature  should  be  held 
constant  for  at  least  an  hour  before  observations  are 
begun.  The  temperature  need  not  fall  below  40°. 
After  the  ordinary  oscillation  determination,  as  shown 
in  the  blank,  the  ice-water  is  drawn  off  and  warm  water 
or  steam  is  passed  through  the  jacket.  The  change  in 
temperature  should  be  gradual,  and  the  higher  tempera- 
ture (maximum  summer-heat  in  the  shade)  should  again 
be  maintained  for  an  hour.  Unless  these  precautions 
are  taken  fallacious  values  of  q  will  result.  If  another 
instrument  is  not  available  for  the  simultaneous  deter- 
mination of  H  (relative  determinations  only  are  needed), 
the  lower  temperature  should  again  be  reproduced  with 
similar  precautions,  in  order  to  eliminate  changes  in  H; 
the  mean  of  the  two  determinations  at  the  mean  of  the 
lower  temperatures  (which  may  differ  two  or  three  de- 
grees) being  combined  with  that  at  the  higher.  Since 
the  magnet  is  oscillated  at  the  two  temperatures,  the 
value  of  /  for  those  temperatures  must  be  used. 
The  equations  for  the  two  temperatures  are  : 


. 

H'MRI 

a' 

Dividing  one  equation  by  the  other,  and  combining 
with  (31), 


from  which 


THEOEY   OF  MAGNETIC   MEASUREMENTS.  59 

If  the  determinations  at  6  and  0'  have  alternated,  as 

TT 

explained  above,  the  ratio  —/may  be  assumed  unity. 

Its  value  may,  however,  be  determined  by  a  deflection 
magnetometer,  which  may  easily  be  extemporized  in 
any  good  laboratory;  and  this  is  strongly  recommended. 
This  instrument  should  be  in  a  room  of  constant  tem- 
perature. The  ratio 

H tan  u' 

H'      tan  u 

if  the  tangent  magnetometer  is  used  [eq.  (38).]  The 
angles  uf  and  u  must,  however,  be  corrected  for  change 
in  declination,  which  may  be  done  by  removing  the  de- 
flecting magnet. 

The  value  of  q  may  also  be  determined  by  the  method 
of  deflections.  It  is  also  best  in  this  method  to  use  a 
second  instrument  to  determine  changes  in  declination. 
The  change  in  H  should  also  receive  a  correction  as  be- 
fore. The  deflection-magnet  is  put  in  position  on  the 
deflection-bar  and  surrounded  with  a  copper  water- 
jacket.  The  deflected  needle  and  declination  are  simul- 
taneously read  at  the  lower  temperature.  If  a  second 
instrument  is  not  available,  the  angle  of  deflection  must 
be  determined  by  removing  the  deflection-magnet  from 
the  bar.  It  is  then  more  difficult  to  control  its  tempe- 
rature, but  with  proper  care  and  patience  the  method 
will  give  good  results. 

If  the  sine  magnetometer  is  used  the  deflection  for- 
mula (39)  gives  the  equation 


hence  mt 

Mfi,  —  M 0      sin  u'  —  sin  u 


sn  u 


(43) 


60  THEORY  OF  MAGNETIC   MEASUREMENTS. 

By  (31) 

M ,  —  M 

—^ — *  =  -  q  (6'  _  8).  (44) 


Since  u'  and  u  differ  very  little  from  each  other, 
sin  u'  —  sin  u  =  cos  u  (u'  —  u)  ;  hence,  by  (43)   and 


If  u'  —  u  =  d,  measured  in  scale  divisions,  and  the 
value  of  one  scale  division  in  minutes  be  s,  the  arc  of  1' 
in  terms  of  radius  being  L,  then  the  above  equation 
becomes 

2  =  cotw  0^70,  (45) 

where  u  is  the  angle  of  deflection  at  the  lower  tempera- 
ture 6. 

If  the  tangent  magnetometer  is  used  equation  (38) 
gives,  as  in  the  previous  case, 


=  _  _ 


MB  tan  u 

If  %'  and  u  are  small  (about  two  degrees),  u  —  u' 
being  not  over  two  or  three  minutes,  which  is  about  its 
usual  value,  the  numerator  tan  u'  —  tan  u  may  be 
written  u'  —  u.  Under  these  conditions  the  expression 
for  q  is  the  same  as  that  deduced  for  the  sine  magnet- 
ometer (45).  In  case  the  above  approximation  is  not 
admissible  the  formula  becomes 

tan  u  —  tan  u' 


The  following  practical  example  will  show  the  method 
of  determination  : 


THEOKY   OF  MAGNETIC   MEASUREMENTS. 


61 


Determination  of  q  for  magnet  <76  of  U.  S.  C.  and  G-. 
Survey,  March  11,  1881.  F.  E.  N.,  observer. 

Magnet  06  deflecting  (717,  which  is  suspended  in  the 
University  magnetometer,  mounted  on  south  pier  of  the 
clock-room  of  Washington  University.  Declinometer 
No.  3,  U.  S.  C.  and  G.  Survey,  with  magnet  No.  1  sus- 
pended, was  mounted  on  north  pier,  in  order  to  correct 
for  hourly  change.  Both  magnets  had  been  suspended 
for  a  week  in  order  to  render  the  fibres  constant,  torsion 
being  corrected.  The  scale  values  of  the  magnets  were, 
(717,  2'. 89 ;  No.  1,  T.902 ;  and  the  scales  are  so  mounted 
that  when  the  easterly  declination  increases,  the  scale- 
reading  of  No.  1  increases,  while  that  of  (717  decreases. 
At  2  o'clock  P.M.  the  scales  read  : 


79.0. 


No.  1,  77.95. 


At  3  P.M.  magnet  C6  was  put  in  place,  deflecting  (7n, 
r  being  twenty-one  inches.  It  was  surrounded  with  a 
copper  jacket  of  ice-water,  which  was  fed  by  a  drip  of 
ice-water  from  a  piece  of  ice  during  the  night.  The 
next  morning  the  scales  read  as  follows : 


SCALE-BEADING. 

Temperature  of 

No.  1. 

017. 

(7.. 

Room. 

8h  30m 

82.0 

149.15 

67.5 

70 

8   45 

81.6 

149.5 

67.4- 

70 

9   40 

80.55 

150.0 

67.5 

70 

10    15 

79.9 

150.2 

67.6 

70 

Mean,  67.5. 

At  10.15  the  ice-water  was  gradually  removed  and 


THEORY   OF  MAGNETIC   MEASUREMENTS. 


hydrant-water  substituted,  which  was  gradually  warmed 
with  a  Bunsen  flame.     The  readings  were  then : 


Hour. 

SCALE-BEADING. 

Temp,  of 

£'«• 

No.  1. 

Cfo 

llh  55m 

77.3 

150.9 

103 

71 

12    10 

77.4 

151.2 

104 

72 

20 

77.4 

151.0 

105 

72 

85 

77.5 

151.0 

104 

72 

50 

77.5 

150.9 

105 

72 

1    00 

77.5 

150.9 

104 

72 

1    17 

77.6 

151.1 

105.3 

72 

Mean, 


104.3. 


At  1.18  P.M.  the  doors  of  the  water-bath  were  opened 
and  G6  was  allowed  to  cool  down  slowly,  the  water  being 
gradually  replaced  by  ice-water  as  before.  The  read- 
ings were  then  : 


SCALE-BEADING. 

Hour. 

Temp,  of 
Ct- 

Temp,  of 
Room. 

No.  1. 

cw 

4hOOm, 

78.3 

151.5 

66 

72 

14, 

78.5 

151.8 

66 

72 

25 

78.65 

151.8 

66.8 

72.2 

36 

78.65 

151.75 

67.5 

72.5 

45 

79.0 

151.7 

67.8 

72.2 

Mean, 


66.8. 


THEOKY   OF   MAGNETIC   MEASUREMENTS. 


63 


At  4.55,  G9  being  removed,  the  suspended  magnets 
read  : 

No.  1,  78.95.  #„,  79.35. 

The  mean  reading  of  No.  1  was  78. 7,  and  the  read- 
ings of  (717  were  corrected  to  this,  as  is  shown  in  the 
following  table.  In  applying  the  correction  to  the 
scale-reading  of  (717  its  sign  is  reversed,  by  reason  of 


Magnet  No.  1. 

Correc- 
tion in 
scale  div. 
of  C17. 

Reading  of  C717. 

Mean 

Reading. 

Correc- 
tion to 
Mean. 

Observed. 

Corrected. 

t-H 

82.0 

-3.3 

+2.2 

149.15 

151.4 

! 

81.6 
80.55 

-2.9 
-1.85 

+  1.9 

+  1.2 

149.5 
150.0 

151.4 
151.2 

151.25 

79.9 

-1.2 

+0.8 

150.2 

151.0 

77.3 

+  1.4 

-0.9 

150.9 

150.0 

77.4 

+  1.3 

-0.86 

151.2 

150.3 

HH 
t—  1 

77.4 

+  1.3 

-0.86 

151.0 

150.2 

1 

77.5 

+  1.2 

-0.8 

151.0 

149.2 

150.04 

1 

77.5 

+  1.2 

-0.8 

150.9 

150.1 

- 

77.5 

+  1.2 

-0.8 

150.9 

150.1 

77.6 

+  1.1 

-0.7 

151.1 

150.4 

78.3 

+  0.4 

-0.26 

151.5 

151.24 

a 

1 

£ 

78.5 
78.65 
78.65 
79.0 

+0.2 
+  0.05 
+  0.05 
-0.3 

-0.13 
-0.03 
-0.03 
+0.19 

151.8 
151.8 
151.75 
151.7 

151.67 
151.77 
151.72 
151.89 

151.66 

64:  THEORY  OF  MAGNETIC  MEASUREMENTS. 

the  fact  that  the  scales  read  in  opposite  directions,  as 
stated. 

For  the  lower  temperature  6,  Series  I.  and  III.,  the 
mean  readings  are  : 

6 

Series  I.,        67.5 
"    III.,        66.8 

Means        67.15  151.46 

For  the  higher  temperature  6', 

6'  Cn  reads 

Series  II.,      104.3  150.04 

Hence  a  change  of  37°.15  F.  in  the  temperature  of 
magnet  Ct  produces  a  change  of  1.42  scale  divisions  in 
the  reading  of  <717. 

The  angle  of  deflection  u  at  the  lower  temperature  is 
determined  from  the  simultaneous  readings  before  and 
after  the  experiments,  as  follows  : 

Before  Series  I.       After  Series  HI. 

No.  1  read 77.95  78.95 

Mean  of  series..  78.7  78.7 


Correction  to  mean -fO.75  —0.25 

Keduced  to  scale  of  <717 —0.49  +0.16 

(717  read 79.0  79.35 


#„  corrected 78.51  79.51 

Cn  during  deflection 151.25  151. 66 

Angle  of  deflection 72. 74  div.        72. 15  div. 

Hence  the  mean  angle  of  deflection  u  at  the  lower 
temperature  is  72.45  scale  divisions  of  (7n,  or  3°  29'.4. 
The  value  of  q  is  therefore  computed  as  follows  : 


THEORY  OF  MAGNETIC   MEASUREMENTS.  65 


V 

5  =  2.89 
d=1.42 
L  =  0.00029 

-0  =  37.15 
o  =  0.00047 

Logs. 

0.461 
0.152 
6.464 
cot.  1.215 
a.c.  8.430 

6.722 

The  observer  should  then  compute  a  table  of  the 
values  of  log  [1  —  (6'  —  6)  q~\  for  values  of  6'  —  6  be- 
tween -f-  9  and  —9,  and  a  table  of  proportional  parts  for 
tenths  of  a  degree,  to  facilitate  the  reduction  of  the 
oscillation  series.  The  value  of  q  depends  on  the 
nature  and  hardness  of  the  steel  of  which  the  magnet 
is  composed.  In  the  above  case  a  previous  determina- 
tion some  years  before  gave  for  this  magnet  a  value 
q  =  0.00048. 


SYSTEMS  OF  UNITS. 

In  the  government  surveys  in  England  and  America 
the  fundamental  units  taken  have  been  the  foot,  the 
grain,  and  the  second.  In  most  scientific  measurements 
the  fundamental  units  used  are  now  the  centimetre, 
gramme,  second.  It  is  therefore  of  some  importance  to 
show  how  the  results  in  one  system  are  to  be  expressed 
in  the  other. 

As  a  preliminary  a  few  simple  illustrations  of  the 
theory  of  physical  units  will  be  given,  in  order  to  make 
the  subject  clear.  For  additional  information  the 
reader  is  referred  to  Everett's  "  Units  and  Physical 
Constants."  If  any  length  is  measured  in  feet  the 
length  may  be  said  to  be  I'  times  the  length  of  one  foot. 
3* 


66  THEORY  OF  MAGNETIC  MEASUREMENTS. 

If  L'  represent  the  length  of  one  foot,  the  whole  dis- 
tance may  be  written  I'L1.  If  the  same  distance  be 
measured  in  centimetres,  I  being  the  number  of  centi- 
metres and  L  the  length  of  one,  then  this  distance  may 
also  be  written  IL.  As  the  distance  is  the  same,  what- 
ever the  system  of  units  used,  it  follows  that 

IL  =  1'L'; 

and  hence 

J  =  |^r.  (47) 

Here  I  and  I'  are  the  numerical  quantities,  which  are 
usually  called  the  "  lengths,"  the  one  being  in  centi- 

Tt 

metres,  the  other  in  feet.    The  ratio  -j-  is  the  length  of 

a  foot  in  centimetres,  or  the  ratio  of  the  dimensions 
of  these  two  units.  By  direct  comparison  this  ratio  has 
been  found  to  be  30.4797.  Any  length  expressed  in 
feet  is  converted  into  the  equivalent  expression  of  the 
same  length  in  centimetres  by  multiplying  by  this  num- 
ber. 
Density  is  defined  to  be  the  mass  per  unit  of  volume 

of  any  body.     It  is  expressed  by  the  ratio  —  ,  where  v  is 

the  volume  of  the  mass  m.  If  M  represents  the  magni- 
tude of  the  unit  mass,  and  L  the  magnitude  of  the  unit 
length,  the  magnitude  of  the  unit  volume  being  then 
Z3,  then  any  density  would  be  completely  represented 
by  the  expression 


If  M  represents  the  magnitude  of  the  gramme,  and 
L  that  of  the  centimetre,  the  unit  density  would  be  a 
gramme  per  cubic  centimetre.  If  M  represents  the 


THEOKY  OF  MAGNETIC   MEASUREMENTS.  67 

kilogramme,  and  L  the  decimetre,  the  unit  density  would 
be  a  kilogramme  per  cubic  decimetre.  The  unit  density 
in  these  two  cases  is  identical,  and  is  that  of  water.  If 
M  represents  the  kilogramme,  and  L  the  centimetre, 
the  unit  density  would  be  a  thousand  grammes  per 
cubic  centimetre,  and  would  be  a  thousand  times  that 
of  water.  The  density  of  water  in  kilogrammes  per 

M 

cubic  centimetre  is  0.001.      Hence  the  expression  -=-8 

Ju 

represents  the  magnitude  or  "  dimensions  "  of  the  unit 
of  density,  in  precisely  the  same  sense  that  M  represents 
the  magnitude  of  the  unit  mass,  and  L  that  of  the  unit 
length. 

If,  then,  D  is  the  density  in  grammes  per  cubic  centi- 
metre, and  D'  the  same  density  in  pounds  per  cubic 
foot,  we  shall  have  an  equation  similar  to  the  one  lead- 
ing to  (47),  viz. : 


or 


If,  for  example,  D  be  taken  as  unity,  which  is  that  of 
water  (grammes  per  cubic  centimetre),  the  equivalent 
density  in  pounds  per  cubic  foot  will  be 

M_  (L'\> 

-M'\L 

Here 

^       gramme  = 
M'        pound 

Keducjng,  the  value  of  D'  is  found  to  be  62.425, 
which  irtlie  density  of  water  in  pounds  per  cubic  foot. 


68  THEOKY   OF   MAGNETIC   MEASUREMENTS. 

Eecurring  now  to  the  equations  for  determining  //, 
viz.: 


M     ,  .* 

ff  =  K  *. 

it  is  required  to  find  the  conversion  factor,  which  will 
reduce  the  value  H  measured  in  foot-grain-second  units 
to  C.  Gr.  S.  units.  It  is  to  be  observed  that  the  value  of 
P  is  so  determined  as  to  satisfy  the  two  equations 

M 

—r  =  ir8  tan  u 

£1 


Solving  these  equations  for  P,  its  value  is  found  to  be 
p      r*  tan  u  —  r"  tan  u' 

r  tan  u  —  r'  tan  u' '    «• 

Since  the  tangents  in  this  expression  are  independent 
of  any  system  of  units,  its  value  can  only  be  changed  by 
a  change  in  the  unit  of  length.  Hence  the  dimensions 

p 
of  P  are  L\    The  ratio  -5-  will  therefore  be  independent 

of  the  system  of  units  used.     It  will  easily  be  seen  that 
the  same  is  true  of  the  torsion  and  temperature  correc- 
tions which  have  been  already  discussed. 
Solving  the  two  general  equations  for  H, 


H=A 

where 


1/5- 


/  27T3 

=v^7^?y 


THEOET  OF  MAGNETIC  MEASUREMENTS.  69 

The  value  of  A  is  independent  of  any  change  in  the 
system  of  units.  /  is  measured  by  a  mass  into  a  dis- 
tance squared,  r*  is  the  cube  of  a  distance.  The  unit 
of  time  is  to  be  the  second  in  both  cases.  Hence  if  H' 
be  the  horizontal  component  of  the  strength  of  the 
earth's  field  measured  in  the  English  units,  and  H  the 
same  strength  in  metric  units  0.  G-.  S., 

B=(*,£fir. 


Substituting  the  values  of  the  unit  ratios, 


log  H=  8.  863778  -flog  H' 
log  H'  =  1.336222  +  log  H. 


EXPLANATION  OF  THE  PLATES. 

Plate  I.  represents  the  U.  S.  Coast  Survey  magnet- 
ometer, having  the  deflection-magnet  in  position  for  ob- 
servation. The  shorter  deflection-magnet  is  suspended 
to  the  torsion-head.  The  instrument  can  be  used  either 
as  a  sine  or  a  tangent  instrument.  It  can  also  be  used 
as  a  declinometer.  In  place  of  the  observing  telescope 
of  I.  a  small  alt-azimuth  instrument  like  that  shown 
in  II.  may  be  used,  being  mounted  on  a  table-tripod  with 
the  magnetometer. 

Plate  III.  represents  the  form  of  dip-circle  now  com- 
monly used.  The  vertical  circle  is  about  six  inches  in 
diameter,  reading  by  opposing  verniers  to  minutes. 
Two  simple'  microscopes  serve  to  read  the  verniers, 


70 


THEORY   OF  MAGNETIC   MEASUREMENTS. 


while  two  compound  microscopes  moving  with  the  ver- 
niers are  pointed  upon  the  marked  ends  of  the  needle. 
These  plates  were  kindly  furnished  by  the  eminent  in- 
strument-makers, Fauth  &  Co.,  of  Washington,  D.  C., 
being  reproduced  from  their  catalogue. 


PLATE  L 


THEOEY  OF  MAGNETIC   MEASUBEMENTS.  71 


PLATE  II. 


72  THEORY   OF  MAGNETIC   MEASUREMENTS. 


PLATE  III. 


APPENDIX 

ON  THE  REDUCTION  OF  OBSERVATIONS  BY 
THE  METHOD  OF  LEAST  SQUARES. 


If  a  blacksmith  were  to  repeatedly  measure  the  length 
of  a  piece  of  iron  by  means  of  his  two-foot  rule,  his  mea- 
surements would  all  agree.  If  the  distance  between  two 
fine  lines  on  the  bar  of  iron  is  measured  with  the  high- 
est attainable  accuracy  by  means  of  a  measuring  engine, 
the  results  of  separate  and  independent  measurements 
do  not  agree  except  by  accident,  and  the  tendency  to 
disagree  is  found  to  increase  with  the  delicacy  of  the 
determination.  It  is  therefore  manifest  that  the  true 
length  of  the  bar  can  never  be  obtained.  It  is  conceded 
universally  that  the  arithmetical  mean  of  the  observed 
values  is  the  best  result  that  can  be  obtained,  when  all 
known  corrections  have  been  made  and  the  observa- 
tions are  all  equally  worthy  of  confidence,  or,  in  other 
words,  when  they  have  equal  weight.  Really  the  obser- 
vations might  be  weighted  differently  by  different  ob- 
servers, if  one  happened  to  notice  something  affecting 
the  value  of  an  observation  which  should  escape  the 
attention  of  the  other.  Unsuspected  causes  may  thus 
affect  consecutive  observations  in  a  different  degree,  so 
that  observations  to  which  one  observer  might  give 
equal  weight  might  be  differently  weighted  by  another. 
It  is  impossible  to  reproduce  exactly  the  same  condi- 


74  APPENDIX. 

tions,  and  to  make  consecutive  observations  in  the  same 
manner.  In  fact,  the  differences  in  the  results  obtained 
are  due  to  such  causes.  When  the  observer  does  not 
know  of  some  specific  reason  for  attaching  less  im- 
portance to  an  observation  than  to  others,  it  should  be 
given  equal  weight,  even  if  it  is  discordant. 


PROPERTIES  OF  THE  ARITHMETICAL  MEAN. 

Let  a  single  quantity  be  measured  n  times,  the 
observation  values  being  xlf  x^  xa,  .  .  .  xn.  Let 
&I+ *«+*•+  •  •  •  vn=^z-  If  ^0  represent  the  arith- 
metical mean,  then 

2x 

—  =  2v  W 

ft 

If  x0  be  subtracted  from  each  of  the  observation 
values,  the  resulting  differences  or  residuals  will  be 
sometimes  positive,  sometimes  negative.  The  smaller 
the  residuals  are  numerically,  the  greater  the  precision 
of  the  measurements,  and  the  greater  the  degree  of  con- 
fidence to  which  the  mean  is  entitled,  provided  all  con- 
stant errors  affecting  all  measurements  alike  have  been 
previously  eliminated.  Indicating  these  residuals  by  r, 
the  n  observations  would  give  residuals  as  follows  : 


etc.,   etc.,  etc., 


By  adding,  2x  —  nx0  =2r  (2) 


THEORY   OF  MAGNETIC   MEASUREMENTS.  75 

By  (1)  it  follows  that  the  first  member  of  (2)  is  equal 
to  zero,  hence 

2r=Q.         .  (3) 

The  arithmetical  mean  renders  the  sum  of  the  resi- 
duals zero.  Any  other  number  thus  treated  would  give 
residuals  the  sum  of  which  would  be  greater  or  less 
than  zero.  If  the  separate  observations  were  all  pre- 
cisely alike,  each  residual  would  be  zero. 

If  the  individual  equations  for  the  residuals  are 
squared,  the  resulting  values  of  ra  are  all  positive.  A 
little  consideration  will  enable  one  to  see  that  the  sum 
of  the  squares  of  the  residuals  obtained  from  the  arith- 
metical mean  will  be  less  than  when  the  residuals  are 
formed  with  any  other  number.  This  is,  in  fact,  easily 
shown.  The  squared  equations  are  : 


etc.,        etc.,        etc., 

rn*  =  x'n*-2x0xn-}-x* 

Adding  these  equations,  the  result  is 

2r9  =  2x*  -  2x0  2x  +  n  XQ\ 

If  any  other  value  K  were  taken  instead  of  the  arith- 
metical mean,  the  residuals  would  have  different  values. 
Let  them  be  called  p.  The  last  equation  would  then 
read 


The  value  of  K  is  to  be  found,  which  will  render 
2p*  a  minimum.  This  value  is  found  from  the  condi- 
tion 

=  —  22x  -f  2nK=  0, 


76  APPENDIX. 

from  which 


If  the  different  observations  are  not  deserving  of 
equal  weight,  the  reduction  must  be  so  made  that  the 
values  deserving  of  most  confidence  shall  have  most  to 
do  with  determining  the  result. 

If  the  variation  of  the  magnetic  needle  were  deter- 
mined by  four  measurements  to  be 

9°  24.1  with  weight  4 
9  25.0  "  "  5 
9  24.8  "  "  5 
9  22.6  "  "  4 

the  best  value  would  be  obtained  by  adding  in  the  first 
and  fourth  values  each  four  times,  the  second  and  third 
each  five  times,  dividing  the  sum  by  the  sum  of  the 
weights,  or  18. 

In  such  a  case  the  weights  of  the  observations  might 
be  determined  by  the  judgment  of  the  observer,  who  is 
able  to  cite  some  specific  reason  for  attaching  less  im- 
portance to  some  observations  than  to  others.  That  the 
result  of  an  observation  is  discordant  is  not  of  itself 
an  important  reason,  and  should  be  accepted  with  very 
great  caution,  although  discordant  observations  will 
really  affect  the  weight  of  the  resulting  mean. 

If  the  numbers  to  be  weighted  are  the  means  of  seve- 
ral equally  good  observations,  their  weights  would  in 
each  case  be  represented  by  the  number  of  observations. 
The  weight  of  such  a  mean  is  also  determined  by  its 
probable  error,  as  is  shown  in  more  extended  treatises, 
in  which  the  theory  of  probability  is  applied  to  errors  of 
observation. 


THEOEY  OF  MAGNETIC   MEASUREMENTS.  77 

OBSERVATIONS  ON  TWO  OR  MORE  QUAN- 
TITIES. 

Let  it  be  assumed  that  observations  have  been  di- 
rectly made  on  three  variable  quantities  u,  v,  y,  which 
some  graphical  or  other  method  has  shown  to  be  related, 
as  is  indicated  in  the  following  equation  : 

y  =  au  -f-  bv.  (4) 

Considering  this  as  representing  a  physical  relation,  a 
and  b  are  constants,  the  true  values  of  which  cannot  be 
determined.  The  value  of  v  might  be  u*,  so  that  the 
second  member  of  the  equation  would  represent  the 
first  two  terms  of  a  series.  It  is  required  to  assign 
values  to  a  and  b  which  will  most  nearly  agree  with  the 
observations  made  on  y,  u,  and  v. 

Each  set  of  simultaneous  observations  will  give  an 
equation  of  the  form 

y  =  w.a-±-v.b,  (5) 

where  y,  u,  and  v  now  become  numerical  quantities  or 
coefficients.  Such  equations  are  called  "  observation 
equations"  or  "equations  of  condition."  The  latter 
term  is  in  more  general  use,  but  the  former  term  seems 
preferable. 

Any  two  of  the  observation  equations  would  deter- 
mine the  values  of  a  and  b.  But,  on  account  of  the  un- 
avoidable errors  of  observation,  some  other  set  of  two 
would  give  somewhat  different  values  of  a  and  b.  The 
values  obtained  from  any  two  would  necessarily  satisfy 
those  equations,  but  they  would  not  in  general  satisfy 
the  other  equations.  In  other  words,  if  for  each  obser- 
vation equation  the  value  y  —  au  —  bv  is  calculated,  as- 
signing to  a  and  b  any  values  determined  as  above,  the 
value  would  not  in  general  be  zero.  As  in  the  similar 


78 


APPENDIX. 


case  where  the  arithmetical  mean  was  treated,   each 
equation  would  take  the  form 


etc.,       etc.,       etc.., 
rn=yn  —  una  —  vnb 

where  the  residuals  would  not  in  general  be  zero  for 
any  values  of  a  and  b  that  could  be  used.  Evidently 
any  values  whatever  might  be  assigned  to  a  and  b,  and 
a  set  of  residuals  would  result.  Values  might  be  cho- 
sen so  that  the  residuals  might  all  be  positive  or  they 


might  all  be  negative.     It  is  evident  that  the  values  of 
a  and  #  which  will  most  nearly  satisfy  all  the  equations 


THEORY   OF   MAGNETIC   MEASUREMENTS.  79 

will  give  small  residuals,  some  of  which  will  be  positive 
and  some  negative.  Without  farther  discussion  it  will 
be  assumed  that  the  best  values  of  a  and  b  will  make  the 
sum  of  the  squares  of  the  residuals  a  minimum.  The 
problem  is  then  to  assign  values  to  a  and  I  which  will 
make  2r*  a  minimum.  The  values  a  and  b  are  then  to 
be  treated  as  independent  variables,  and  with  the  vari- 
able 2v*  they  determine  a  surface.  The  conditions  for 
the  minimum  point  on  this  surface  are: 


The  first  equation  is  the  condition  for  the  minimum 
point  in  any  section  of  the  surface  parallel  to  the  plane 
determined  by  the  axes  of  2r*  and  b.  The  second  is  the 
condition  for  minimum  on  any  section  at  right  angles 
thereto.  If  the  two  conditions  are  simultaneously  im- 
posed it  determines  a  minimum  point  in  the  surface. 
To  find  the  value  of  -5V3,  the  residuals  are  squared, 
which  gives  the  equations  : 

**if  = 

y  '  -  2u^  .  a  -  2V&  .  J  +  w/  .  a'  +  %uj),  .  ab  +  v?  .  b* 


y'~  Zu,y,  .  a  -  2v,y,  .b  +  u,'.a*  +  2u,v,  .  ab  +  v,1  .  b* 

etc. 
= 

y*  —  2unyn  .  a  —  2vnyn  .b  +  un\a*  +  2unvn  .  ab  +  vna  .  b* 

Adding  these  equations,  the  result  is 


(6) 


80  APPENDIX. 

This  is  the  equation  of  the  surface.  The  intersection 
of  this  surface  by  the  plane  of  the  axes  2v*  and  I  is  de- 
termined by  introducing  into  it  the  condition  a—Q. 
Ita  equation  is  therefore 


which  is  the  equation  of  a  conic  section.  Making 
5  =  0  in  this  equation,  the  value  of  the  intercept  on  the 
axis  2r*  is  2y*.  The  minimum  point  on  this  section 
is  given  by 


from  which 

2vy 


The  sign  of  b  will  here  depend  upon  the  sign  of  the 
numerator. 

It  is  not  necessary  to  further  discuss  this  surface,  aa 
it  is  only  intended  to  point  out  the  nature  of  the  relation 
with  which  we  are  dealing. 

The  conditions  for  the  minimum  point  on  tho  surface 
are  obtained  from  (6).  They  are  : 


The  first  equation  determines  a  minimum  point  on 
any  section  at  right  angles  to  the  b  axis.  The  second 
determines  a  minimum  point  on  any  section  at  right 
angles  to  the  a  axis.  If  the  two  equations  are  com- 
bined by  elimination,  the  values  of  a  and  b  may  be  ob- 
tained, which  determine  the  minimum  point  on  the 
surface. 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


81 


These  values  are : 

2v*  2  uy  —  2uv  2  vy 


_ 


uy 


—  ^  uv  ^  uv.  J 


(8) 


The  calculations  are  made  as  is  shown  in  the  sub- 
joined table,  in  which  the  observed  quantities  are  given 
in  the  first  three  columns,  the  quantities  called  for  by 
equations  (8)  being  the  sums  of  the  succeeding  columns. 


No. 


u* 


uy 


vy 


2  uv 


Equations  (7)  are  called  "normal  equatiom"  the  first 
being  called  the  normal  equation  for  a,  and  the  second 
the  normal  equation  for  b.  It  will^be  observed  that 
they  may  be  obtained  from  the  n  observation  equations 
(5)  as  follows  :  To  find  the  normal  equation  for  a,  each 
observation  equation  is  multiplied  through  by  the  co- 
efficient of  a  in  that  equation.  The  sum  of  the  result- 
ing equations  is  the  normal  equation  for  a.  Similarly 
the  normal  equation  for  1)  is  obtained  by  using  the  co- 


82  APPENDIX. 

efficients  of  b  as  multipliers,  as  will  be  seen  by  inspect- 
ing (5)  and  (7). 

This  process  is  really  equivalent  to  giving  weights  to 
each  equation  equal  to  the  coefficient  of  the  quantity 
whose  normal  equation  is  to-  be  formed.  These  weighted 
equations  are  then  added  together,  giving  a  single  equa- 
tion. If  this  equation  were  divided  through  by  n  it 
would  give  a  properly  weighted  mean  equation.  This 
division  by  n  is,  of  course,  unnecessary  in  solving  the 
equations  for  a  and  b,  as  it  would  not  affect  their  values. 


WEIGHTED  OBSERVATIONS. 

If  the  observations  are  not  of  equal  weight  they 
must  be  additionally  weighted.  If  y#  ult  vy  are  each 
the  mean  of  pv  equally  good  observations,  y^  u^  vy  the 
means  of  p^.  observations,  etc*,  the  normal  equations  be- 
come, as  will  be  easily  seen, 


(9) 


^—  =  b  2pv*  -f  a  2puv  —  2  pvy  =  0 
By  elimination  the  values  of  a  and  t  are : 


"i 
~~  pv*  —  2puv  2puv 


*  2  pvy  —  2puv2puy  I 
2puv  J 


The  computations  are  made  exactly  as  in  the  preced- 
ing case,  excepting  that  an  additional  column  is  added 


THEORY  OF   MAGNETIC   MEASUREMENTS.  83 

just  after  the  first,  containing  the  weight  assigned  to 
each  set  of  values  u,  v,  y.  These  weights  may  be  as- 
signed wholly  on  the  judgment  of  the  observer,  if  this 
is  possible,  or  they  may  simply  be  the  number  of  obser- 
vations of  which  each  is  the  mean,  modified,  if  deemed 
proper,  as  the  judgment  of  the  observer  may  decide. 

Where  each  set  of  values  u,  v,  y  is  the  mean  of  a  large 
number  of  observations,  the  theory  of  probabilities 
enables  one  to  calculate  the  weights  from  the  probable 
errors,  but  ordinarily  this  is  a  wholly  useless  refine- 
ment. 

If  the  original  function  has  the  form 

y  =  a  +  bv,  (11) 

the  value  of  u  in  (4)  becomes  unity.  (In  fact,  (4)  may 
be  put  into  this  form  by  dividing  through  by  u.  The 

11  v 

quantity  -  would  then  replace  y,  and  the  quantity  — 

would  replace  v  in  all  subsequent  equations,  and  u 
would  be  replaced  by  unity.) 

Putting  u  =  l  or  2u*  =n  in  (8),  the  equations  be- 
come 


a= 


n  2  vy  —  2v  2y 


(13) 


The  manner  of  calculating  is  apparent  from  the  table 
following  equation  (8). 
If  the  original  function  has  the  form 

y  =  lv,  (13) 

the  value  of  a  becomes  zero.     There  can  be  no  normal 
equation  for  «,  and  the  first  of  equations  (7)  must  dis- 


84  APPENDIX. 

appear.     In  the  normal  equation  for  b  in  (7)  the  condi- 
tion a  — 0  must  also  be  introduced. 
The  value  of  b  becomes  then 

<"> 


It  will  be  observed  that  this  result  is  different  from 
that  obtained  by  taking  the  average  values  of  the  quan- 
tity -,  which  would  be  -  2  -.  This  is  simply  due  to 

J  v  n      v 

the  fact  that  (14)  is  the  result  of  weighting  the  observa- 
tion equations  in  proportion  to  the  magnitude  of  the 
observed  quantities  or  coefficients  y  and  r  in  the  various 
equations. 

Finally,  if  observations  are  made  on  a  single  quantity, 
so  that  the  function  has  the  form 

y  =  l),  (15) 

the  value  of  v  in  (13)  and  (14)  becomes  unity  and 
2v*  =  n.  Hence  (14)  becomes 

J=?'  (16> 

or  the  best  value  of  the  quantity  ft  is  the  arithmetical 
mean  of  the  observations — a  result  which  has  already 
been  agreed  upon  under  that  head  in  the  early  part  of 
the  discussion. 

The  additionally  weighted  equations  corresponding  to 
(12)  are 

2  pv*  2 py — ^2pv^2pvy  "] 

PV    \  (17) 

2sp  2  pvy  —  2pv  2 py 


THEOKY   OF   MAGNETIC   MEASUREMENTS.  85 

For  (14)  weighted  equations  give  the  value  of  b, 


and  finally,  for  weightedL  observations  on  a  single  quan 
tity,  (18)  becomes 


as  was  shown  in  the  special  example  given  on  p.  76. 


GRAPHICAL  METHODS. 

The  preliminary  investigation  of  any  new  function  is 
always  made  by  graphical  methods.  In  a  majority  of 
cases  met  in  practice  the  graphical  method  is  sufficient. 
In  order  to  be  able  to  make  use  of  this  method,  the 
computer  must  be  familiar  with  the  analytical  geome- 
try, so  that,  from  the  curve  which  is  obtained  by  plotting 
the  observed  values  of  the  variables,  he  can  form  an 
idea  of  the  mathematical  relation  sought.  By  far  the 
greater  number  of  cases  which  are  met  in  physical  in- 
vestigation are  represented  by  the  equation 

y  =  ~bxn. 

Here  y  and  x  are  the  physical  variables,  and  b  and  n 
are  unknown  constants,  the  values  of  which  are  to  be 
determined  so  as  to  best  satisfy  the  equations.  If  n  —  0, 
then  y  =  b,  or  the  function  is  that  given  in  (15).  If 
n  =  1,  then  y  is  directly  proportional  to  x,  and  the 
plotted  values  of  y  and  x  will  give  a  straight  line  pass- 
ing through  the  origin.  If  n=  —  1,  the  equation  is  an 
inverse  proportion,  and  the  curve  will  be  an  equilateral 


86 


APPENDIX. 


hyperbola.     If  n  =  2  or  £,  the  curve  will  be  a  parabola, 
etc. 

In  some  cases  it  may  be  necessary  to  add  a  con- 
stant term  to  the  second  member,  so  that  the  equa- 
tion will  take  the  form  of  (11).  To  take  a  special 
case,  in  order  to  make  the  manner  of  reduction  well 
understood : 


Days. 

log  M. 

y 

d 

0 

9.83989 

-  0.00260 

+   134 

6 

990 

-    261 

+   128 

20 

990 

-    261 

+   114 

23 

9.84055 

-    326 

+   HI 

35 

9.83894 

-    165 

+    99 

50 

762 

-     33 

+   84 

60 

868 

-    139 

+    74 

64 

846 

-    117 

+    70 

74 

901 

-    172 

+    60 

86 

766 

-     37 

+   48 

106 

728 

+      1 

+    28 

129 

778 

-     49 

+    5 

146 

622 

4-    107 

-   12 

156 

714 

+     15 

-   22 

165 

784 

-     55 

-   31 

169 

700 

+     29 

-   35 

171 

623 

4-    106 

-   37 

184 

656 

+     73 

-   50 

195 

679 

+     50 

-    61 

202 

504 

+    225 

-   68 

218 

680 

+     49 

-   84 

228 

568 

+    161 

-    94 

234 

451 

+    278 

-   100 

245 

437 

+    292 

-   Ill 

373 

242 

+    487 

-   239 

Means:  134     9.83729 

In  1865-6  Professor  Wm.  Harkness  made  a  series  of 
intensity  determinations,  and  deduced  the  log.  moment 
of  his  magnet  at  the  several  temperatures  of  observa- 


THEOEY   OF  MAGNETIC   MEASUREMENTS.  87 

tion.*  These  values  were  reduced  to  the  mean  tempe- 
rature of  his  series  by  means  of  equation  (31).  The  re- 
sults are  given  in  the  annexed  table,  where  the  first 
column  gives  the  number  of  days  from  his  first  observa- 
tion, on  Oct.  24, 1865,  and  the  second  column  the  value 
of  log  M  at  a  temperature  75°.  8  F. 

The  values  in  the  two  columns  being  plotted,  the 
points  thus  determined  are  shown  on  the  diagram  (p.  89), 
It  is  manifest  that  if  any  assumption  regarding  the  de- 
crease in  log  M  be  made,  it  must  be  that  of  uniform  de- 
crease. The  equation  representing  this  relation  will  be 

log  M  =  log  M0  —  ad, 

where  log  MQ  is  the  value  of  log  M  at  any  assumed  date, 
and  d  is  the  number  of  days  from  the  assumed  date  to 
that  of  any  other  observation,  a  being  the  daily  change 
in  the  value  of  log  M.  If  the  mean  of  all  the  values  of 
log  M  be  taken  as  log  Jf0,  it  gives  the  value  of  the  quan- 
tity log  M  for  the  mean  date  of  the  series,  which  is  134 
days  after  the  first  observation.  The  straight  line  rep- 
resenting the  observations  must  run  through  the  point 
determined  by  these  two  mean  values.  This  line  is  also 
to  be  so  drawn  as  to  give  weight  to  other  points  in  pro- 
portion to  their  distance  from  the  point  representing 
the  mean  values.  The  equations  of  condition  become 
of  the  form 

log  M0  —  log  M—  ad  =  0, 

where  log  J/"0  =  9.83729,  and  where  d  is  estimated  in 
days  from  the  134th  day,  which  corresponds  to  March 
17',  1866. 

*  "  Smithsonian  Contributions  to  Knowledge,"  vol.  xviii.  p.  55 
of  his  memoir. 


88  APPENDIX. 

Calling  log  M0  —  log  M=y,  the  equations  of  condi- 
tion become 

y  —  ad  =  0. 

'  These  values  of  y  and  d  are  given  in  the  third  and 
fourth  columns  of  the  table.  In  order  to  form  the  nor- 
mal equation  for  a,  each  observation  equation,  of  which 
there  are  twenty-five,  is  multiplied  through  by  d,  the 
coefficient  of  a,  and  the  resulting  equations  are  added. 
The  normal  equation  becomes 


Performing  the  calculations,  the  value  of  u  will  be 
found  to  be 


and  hence  the  original  equation  becomes 

log  M  =  9.83729  -  0.0000195  d. 

It  is  evident  that,  after  having  plotted  the  values  of 
log  J/and  d,  the  position  of  the  line  can  be  determined 
with  a  precision  sufficient  for  most  purposes  by  means 
of  a  fine  thread,  which  is  laid  through  the  points  in 
such  a  way  as  to  agree  with  them  as  nearly  as  possible. 
The  position  of  this  line  is  shown  on  the  diagram. 
If  desired,  one  point  on  the  line  may  be  determined 
with  precision  by  obtaining  the  means  of  the  observa- 
tions as  in  the  first  two  columns  of  the  table.  After  the 
line  is  drawn  the  slope  of  the  line,  or  the  value  of  («), 
is  then  found  by  measuring  on  the  diagram  the  co-ordi- 
nates x',  y'  and  x",  y"  of  any  two  points,  which  should, 
of  course,  be  as  far  apart  as  possible.  In  this  case 

y'  -  y" 

a=2-n  —  £7. 

x"  —  x 


THEORY   OF   MAGNETIC   MEASUREMENTS. 


89 


Mean 


/\ 


Date 


0 


© 
P 


90  APPENDIX. 

The  co-ordinates  are,  of  course,  to  be  measured  in 
terms  of  the  scales  used  in  plotting. 

Such  graphical  solutions  are  in  the  large  majority  of 
cases  sufficient  for  all  purposes.  They  should  in  all 
cases  precede  any  more  exact  mathematical  solution,  in 
order  that  one  may  see  whether  the  observations  are 
sufficiently  precise  to  warrant  a  more  exact  solution,  or 
whether  the  assumed  equation  is  in  harmony  with  the 
observations. 


TIME  OF  ELONGATION  OF  POLARIS. 

EXPLANATION   OF  TABLES   I.,    II.,    AND   III. 

The  following  tables  (pp.  92-93)  give  the  astronomi- 
cal times  of  elongation  of  the  Pole  star  for  the  1st  and 
15th  of  each  month  from  1885  to  1895.  They  are  com- 
puted for  a  latitude  of  40°  and  a  longitude  of  6hrs  from 
Greenwich.  From  them  the  local  astronomical  time  of 
elongation  to  the  nearest  minute,  for  any  latitude  be- 
tween 25°  and  55°,  may  be  obtained  by  applying  the 
correction  given  in  Table  II.  The  correction  for  differ- 
ence of  longitude  is  insignificant,  amounting  to  0'M5 
for  a  difference  of  one  hour,  to  be  subtracted  for  places 
west  of  the  6th  meridian,  and  added  for  places  east. 

To  obtain  the  time  of  elongation  for  any  date  not 
given  in  the  tables,  subtract  3m.94  from  the  tabular  time 
of  elongation  for  every  day  elapsed,  if  the  tabular  date 
is  the  smaller,  or  add  the  same  correction  if  the  tabular 
date  is  the  larger.  This  correction  may  be  obtained 
from  Table  III. 

The  astronomical  day  begins  at  noon,  and  is  twelve 
hours  behind  the  civil  date.  From  noon  to  midnight 


THEOKY   OF   MAGNETIC   MEASUEEMENTS.  91 

the  astronomical  and  civil  dates  are  the  same  ;  from 
midnight  to  noon  the  civil  date  is  one  greater.  Thus 
Jan.  12,  14h  40m,  astronomical  time  is  Jan.  13,  2h  40m 
A.M.  civil  time. 

Example — Eequired  the  time  of  eastern  elongation  of 
Polaris  on  Aug.  8,  1888,  for  a  place  whose  latitude  is 
44°  30'. 

Time  of  elongation,  1888,  Aug.  1  (by  Table  I.)..  10h  38m 
Correction  for  latitude         (by  Table  II.) . .      —1.2 
"   7  days  (by  Table  III.)       -27.6 

Time  of  elongation  Aug.  8,  1888 10h  9ra 


APPENDIX. 


TABLE  I.— EASTERN  ELONGATIONS. 


188C. 

1887. 

1888. 

1889. 

1890. 

1891. 

1892. 

1893. 

1894. 

1895. 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

h  in 

h  m 

h  m 

Apr.  1. 

1838 

1839 

1836 

1838 

1839 

1840 

1837 

1838 

1839 

1840 

"  15. 

1743 

1744 

1741 

1743 

1744 

1745 

1742 

1743 

1744 

1745 

Mayl. 

1640 

1641 

1638 

1640 

1641 

1642 

1639 

1640 

1641 

1643 

"  15. 

1545 

1546 

1544 

1545 

1546 

1547 

1544 

1545 

1546 

1548 

June  1. 

1438 

1440 

1437 

1438 

1440 

1441 

1438 

1439 

1440 

1441 

"  15. 

1344 

1345 

1342 

1343 

1345 

1346 

1343 

1344 

1345 

1346 

July  1. 

1241 

1242 

1239 

1241 

1242 

1243 

1240 

1241 

1242 

1244 

"  15. 

1146 

1147 

1145 

1146 

1147 

1148 

1145 

1146 

1147 

1148 

Aug.  1. 

1040 

1041 

1038 

1039 

1041 

1042 

1039 

1040 

1041 

1042 

"  15. 

945 

946 

943 

944 

946 

947 

944 

945 

946 

947 

Sept.  1. 

838 

839 

836 

838 

839 

840 

837 

838 

839 

841 

"  15. 

743 

744 

742 

743 

744 

745 

742 

743 

744 

746 

Oct.  1. 

640 

641 

639 

640 

641 

642 

639 

641 

642 

643 

WESTERN  ELONGATIONS. 


1886-7 

1887-8 

1888-9 

1889-90 

1890-1 

1891-2 

1892-3 

1893-4 

1894-5 

hm 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

h  m 

Oct.  1. 

1830 

1831 

1828 

1829 

1831 

1832 

1829 

1830 

1831 

"  15. 

1735 

1736 

1733 

1734 

1736 

1737 

1734 

1735 

1736 

Nov  1. 

1628 

1629 

1626 

1628 

1629 

1630 

1627 

1628 

1629 

"  15. 

1533 

1534 

1531 

1532 

1534 

1535 

1532 

1533 

1534 

Dec.  1. 

1430 

1431 

1428 

1429 

1431 

1432 

1429 

1430 

1431 

"  15. 

1335 

1336 

1333 

1334 

1336 

1336 

1334 

1335 

1336 

Jan.  1. 

1227 

1229 

1226 

1227 

1228 

1229 

1226 

1227 

1229 

"  15. 

1132 

1133 

1131 

1132 

1133 

1134 

1131 

1132 

1133 

Feb.  1. 

1025 

1026 

1023 

1025 

1026 

1027 

1024 

1025 

1026 

"  15. 

930 

931 

928 

929 

930 

932 

929 

930 

931 

Mar.  1. 

835 

832 

833 

834 

835 

832 

833 

835 

836 

"  15. 

739 

736 

738 

739 

740 

737 

738 

739 

741 

Apr.  1. 

632 

630 

631 

632 

633 

630 

631 

633 

634 

THEOltY   OF   MAGNETIC   MEASUREMENTS. 


93 


TABLE  II. 

Latitude. 

Correction. 

25° 

+lm.9 

28 

+  1    .6 

31 

+  1  .2 

4 

+0  .8 

37 

+  0  .4 

40 

+0  .0 

43 

-0  .5 

46 

-1  .0 

49 

-1  .6 

52 

-2  .3 

55 

-3  .0 

TABLE  III. 

No.  days. 

Correction. 

1 

3.9 

2 

7.9 

3 

11.8 

4 

15.8 

5 

19.7 

6 

23.6 

7 

27.6 

8 

31.5 

9 

35.5 

10 

39.4 

11 

43.3 

APPENDIX. 


1 

q 
o 

e 
£ 

q 

1 
p 

g 

£H 
O 

o 

q     i>  TJ;  i^  o»  oo  »  o^o  o  o  oq  q  <N  o  05  r*  os »« i^  as  oq  oq  oq  q  »  oo 

I 

q  °.  J>  to  TO  T*  q  05  os  q  q  o*  -^  cq  os  TO  oq  TO  os  to  n<  TO  «s«  TO  «5  oq  so 

-  *  : :.' 

o 


